Materials science for biomedical engineeringEvelinda Baerends. Of course, all errors that are undoubtedly still in this book are my responsibility. Second edition Marcel van Genderen

Materials science for biomedical engineering

Citation for published version (APA):Genderen, van, M. H. P. (2008). Materials science for biomedical engineering. (2nd ed. ed.) Eindhoven:Technische Universiteit Eindhoven.

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F 0 R B I 0 M E D 1-;CÄ L ;.

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Materials Science

for

Biomedical Engineering

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Materials Science for Biomedical Engineering

2nd edition

Marcel van Genderen

Department of Biomedical Engineering

Eindhoven University of Technology

The Netherlands

Cover design and printing by Jan-Willem Luiten, JWL Producties

Copyright© 2008 Department of Biomedical Engineering, Eindhoven Univarsity of Technology, The Netherlands

ISBN 978-90-386-1295-9

For Marie-Thérèse

Acknowledgements First edition Th is book was written over a number of years. lt starled when I took over the Materials Science class, and looked for an appropriate book for Biomedical Engineering students. There wasn't one. Then I decided on a reasonable book, to which I added notes on the topics I thought were missing. This turned out to be quite confusing for the students. so I starled working on expanding my notes with the parls of the book that I needed. At that point, Berl Meijer said: "You should write a book", and it is now finished because he gave me the opportunity to do it. Thanks for that, Bert. Also, credit should go to Fons Sauren for continuously (but gently) pushing me to finish it. Leon Govaerl gave me a good idea for a way to structure the whole class. The terrific books of J.E. Gordon (see Appendix D) helped me a lot: they made me understand what it was all about. The Biomedical Engineering students during several years must be thanked for critica! reading of the various incarnations of my notes, and for trying to understand what I wanted to teil them. Several students helped me with proofreading and making the figures, especially Jeroen Hamers and Evelinda Baerends. Of course, all errors that are undoubtedly still in this book are my responsibility.

Second edition

Marcel van Genderen October 2005

For the second edition, many errors, inconsistencies and unclear passages were found by the Biomedical Engineering students using this book: many thanks to them! Especially Bart Spronck a_nd Stijn Aper were very helpful. Also, some of the symbols have been changed to agree with the new Biomechanics book of Oomens, Brakelmans and Baajjens. Any mistakes that still remain in this edition are of course completely due to me.

Marcel van Genderen May 2008

Contents

Before we start

Part 1: General

1 Introduetion 1 .1 Length scales 1.2 Materials classification

2 Mechanica! properties 2.1 Elastic behaviour 2.2 Tensile test plastic behaviour, fracture 2.3 True stress and strain 2.4 Overview of material properties Problems

3 Bands between atoms: Microscopie structure and elastic modulus 3.1 Bond energy 3.2 Farces on bands 3.3 General characteristics Problems

Part 11: lnorganic Materials

4 lnorganic Materials 1:

1

3 4 4

7 8 13 18 22 23

27 28 31 33 34

Structure of crystalline metals and ceramics, and glasses 35 4.1 Introduetion 36 4.2 Crystallattices 38 4.3 Cubic and hexagonal structures 42 4.4 Polymorphism and polycrystallinity 47 4.5 Milier indices 49 4.6 Glassas 54 Problems 56

5 lnorganic Materials 11: Mechanica! properties and structural defects 59 5.1 Yield and defects 60 5.2 Brittie fracture and crack growth 68 5.3 Glasses 78 Problems 79

viii Materlafs Science for Biomedical Engineering

Part 111: Organic Materials

6 Organiç Material& 1: Structure of (bio )polymers 6.1 Overview 6.2 Morphology: crystalline/amorphous 6.3 Biopolymers Problems

7 Organic Materials 11: Mechanica! properties 7.1 Overview 7.2 Stiffness 7.3 Visco-elastic behaviour 7.4 Rubber elasticity 7.5 Fracture 7.6 Machanical properties of biopolymers Problems

Part IV: Changing Materials

8 Changing material properties 1: Microstructure 8.1 Changing metal properties 8.2 Phase diagrams 8.3 Polymers Problems

9 Changing material properties 11: (Bio}composites 9.1 Structure 9.2 Machanical properties 9.3 Biocomposites Problems

81 82 89 96 101

103 104 106 109 119 121 122 125

127 128 132 145 146

151 152 152 158 168

Contents ix

Appendices 171

Appendix A: English-Dutch glossary 172 List of symbols 180

Appendix B: More on true stress and strain 185 Appendix C: Examples of towers 188 Appendix D: Literature 191 Appendix E: Selected material properties 192

Table 1. Modulus of elasticity, shear modulus, Poisson ratio, density, and reduced modulus of various materials at room temperature 192

Table 2. Typical values for plastic deformation of metals at room temperature 193

Table 3. Slip systems and critica! shearing stress in crystals 193 Table 4. Machanical properties of polymers 194 Table 5. Toughness of some materials 195 Table 6. Machanical properties of metals 196 Table 7. Properties of reinforcing fibres, matrices, and

composites 197 Appendix F: Non-SI units 198 Appendix G: Periadie table; Physical constants 199

Properties of elements 200 Appendix H: Rubber elasticity 207 Appendix 1: Polymer structures 209 Appendix J: Phase rule 211 Appendix K: Lever rule 212 Appendix L: Packing in ionic solids 213 Appendix M: The Orowan model for brittie fracture 214 Appendix N: The random-coil model in three dimensions 216 Appendix 0: Answers to Problems 217

Index 222

x Materlafs Science for Biomedicaf Engineering

All our science, measured against reality, is primitive and childlikeand yet it is the most precious thing we have.

Before we start

Albert Einstein (1879-1955)

Why do biomedical engineersneed to know about materials science? Mostly, books about materials science are concerned with constructing bridges or building planes, and involve the use of concrete and steel. Newer applications of materials science are found in the solid-state devices for micro-electronics, where one needs to know how to work with semiconductors, or the development of advanced polymerie matenals for a variety of purposes.

From an engineering viewpoint, the human body is a (magnificently complex) construction with an advanced control system built-in. lf we want to understand this construction, how it is built, how it behaves, and especially if we want to modify, repair, or imprave it, we need to know sarnething about the basics of the materials that are found in the human body and of the materials that we want to put in there. We will see that we can use a lot of knowledge that machanical engineers have acquired, but that we will also needother areasof science (chemistry, biochemistry, biology) in order to understand the human body from a machanical engineering viewpoint

Generallnformatlon

In this book we wiJllook at the relation between the structure of materials and their mechanica! properties. This needs partsof various scientific disciplines: chemistry, physics, mechanics, and biochemistry. Unfortunately this information was up to now nat available in one textbook. The present book is based on saveral books that are worth studying by themselves for further explorations in materia Is science (listed in Appendix D).

Words that are printed bold (except this one, of course) are lisled in the Glossary (Appendix A). Usually, bold is only used in the section where the term is defined or discussed in most detail.

In formulas, units of quantities will be given between square brackets if necessary to avoid misunderstanding, e.g. energy: E [J].

2 Materials Science for Biomedical Engineering

Overview of the book

As mention9d in the previous section, the main objective of this book is to show the relation between the structure of a material and its mechanica! properties. We will see that we need to look at the structure, not only at the level of atoms and bands (microscale), but also on a larger scale where we can see the imperfectionsin the material (mesoscale). As engineers, you must be able not only to explain the relation between structure and properties, but also to perfarm calculations on several aspects of bath structures and properties.

After an introduetion to materials and the definition of various terms in Chapter 1, we will first take a look at the relevant machanical properties (Chapter 2). Then, in Chapter 3, we willlook at the only mechanica! property that can be derived from the atomie (microscale) structure: the modulus of elasticity.

In the rest of the book, we need to look at both the micro- and mesaseale structure in order to understand the machanical (macroscale) properties. Therefore, there will always be a Chapter on structure, foliowed by a Chapter on properties.

First, we will look at the matenals that are based on strong covalent or ionic bonding, and that usually farm crystalline structures: the inorganic materials (Chapters 4 and 5).

Then, we will study the materials that are based on macromolecules with weak intermolecular bands: the polymers and biopolymers (Chapters 6 and 7), which are usually partly crystalline and partly amorphous.

In the final two Chapters, we willlook at ways to change the properties of materials: various processing techniques to change the structure (Chapter 8), and the making of composita matenals (Chapter 9). Also, we will take a first look at biocomposites, which occur a lot in the hu man body.

Most Chapters have problems associated with them, and answers are given in Appendix 0. Problems indicated with *are typical for the degree of difficulty to be expected in examinations. There are also various Appendices with important data that you will need for making the problems, sa take a look at them.

There are various lntermezzos in the text: these are extra, and do not belang to the material that will be tested.

1 Introduetion

Materials are in general all solids, so this book will deal with various kinds of solid materials.

Materials can be looked at from various perspectives, and in this chapter we will take a look at the ones that are most useful to US.

By doubting we come at truth.

Marcus Tullius Cicero (106-43 BCE)

4 Matenals Science tor Biomedical Engineering

1.1 Length scales

Materialsof course have many types of properties (electrical, magnetic, optica!, thermal ... ), but in general it is possible to assign them to typ i cal classes: metals, (bio)ceramics (and glasses), and (bio)polymers. Here, we will concentrale on the relation between the mechanica! properties of a material and its structure. Mechanica! properties will include elasticity and strength in terms of lension and compression, twisting, shearing, and bending.

In order to understand the relation between the mechanica! properties of a material (macroscale: a length scale of> 1 mm) and its structure, we will see that we need to look at the structure not only at the level of atoms and bonds (microscale: length scale of ca. 1 nm), but also on a larger scale where we can see the imperfectionsin the material (mesoscale: length scale of ca. 1 1-1m). This is illustrated in Figure 1.1.

.. ..

Figure 1.1. From macroscale to mesaseale to microscale.

1 .2 Materials classification

There are saveral ways to classify materials, but not all are useful for us here: e.g. the chronological classification (wood and bone, stone, copper, bronze, iron, steel, plastics). In the following paragraphs we willlook at some important classifications and aspects of materials.

1.2.1 Natura/ vs. synthetic materials

Nature uses a rather restricted set of building blocks tor materials. Synthetic matenals in general are stronger and stiffer than natura I materials, but they a lso have a higher density. Composites can combine the best of these properties. Natural materials have a highly optimized construction compared to synthetic materials, they can grow in place inslead of being manufactured, and they can be repaired.

1 Introduetion 5

1.2.2 Engineering materfals vs. biologica/ materfals vs. biomaterials

Most books about materials treat the engineering materials (steel, concrete): they are used to build briqges and airplanes. In this book we willlimit ourselves mostly to: (i) synthetic materials that can be used in the human body. These are by definition biomaterials: polymers, such as nylon, silicones, Teflon and Dacron; metals, such as titanium, steel, Co/Cr-alloys and gold; ceramics, such as alumina (AI203), carbon, and hydroxyapatite; and composites; (ii) biologica! materials that are in the human body (proteins, polysaccharides, bioceramics), or that can be used as biomaterials in a modified form (spider silk, collagen).

Biologica! materials can be very complex: tissues, muscles, extracellular matrix, etc. We will only be able to look at them briefly in this book.

1.2.3 Simple materfals vs. composites & structures

Monolithic materials (whole structure is one material) are much simplar to understand, but composites (combinations of materials) are more relevant for modern and biologica! materials. Therefore we will start with monolithic materials, but move to composites later on in the book.

Hierarchical and cellular materials or structures-the difference is less clear here-are especially important in biologica! systems: there are various levels of construction, which all influence the material's properties.

Examples of hierarchical systems are tendon (Figure 1.2) and bone, while bone also is a composita materiaL

tropocollagen 1.5 nm

subfibril 10-20 nm

microfibril 3.5nm

fibril 50-500 nm faseiele

50-300 J.lffi

tendon 1 00-500 J.lffi

Figure 1.2. A hierarchical structure occurring in the human body.

Cellular materials or structures are built up from open spaces and thin walls, and are found both in nature (such as cork) and in synthetic materials (think of bubble plastic).

6 Materials Science for Biomedica/ Engineering

1.2.4 Types of chemica/ bonds

Atoms or molecules have to stick logether to form materials. This can occur via strong chemica! bonds, or via weaker secondary interactions. The strong bonds are either based on electrastatic attraction between oppositely charged ions, or on shared electrons in covalent bonds.

The secondary interactions are all based on permanent or transient dipoles. The weakest interaction is the van der Waals attraction, which is always present, even between completely apolar molecules. A stronger attraction is caused by dipoledipole interactions between polar molecules. The presence of an O-H or N-H group leads to hydrogen bonds, which are the strengest secondary interactions.

Atoms can form bonds via electrostatle (Coulomb) interactions when they lose or gain electrans to form ions. This leads almost always to crystalline compounds (salts) where the cations and anions are stacked in a regular lattice. Salts are a form of caramie materials.

Atoms can also form covalent bonds by overlap of orbitals. This can lead to: (i) the formation of small or large molecules (see below), or (ii) the formation of a covalent network, which is already a material (e.g., diamond; silicon). These latter materials are also ceramics.

A special kind of covalent network is seen for the metals: they do not form individu al bonds between atoms, but form one large molecular orbital tor all atoms in a piece of material: the metallic bond or electron "sea".

Small and large molecules can form a material via intermolecular interactions: van der Waals, dipole-dipole, hydragen bonding and electrostatle interactions. Since these secondary interactions are rather weak, this will not lead to useful materials, unless the molecules are very large (macromolecules) and there are many interactlans that add up. These materials are the (bio)polymers.

1.2.5 Biomedical aspects

For biomaterials there are a number of aspects that are not important for the normal engineering materials: biocompatibility, sterilization, cell and protein adsorption, biodegradation/erosion, bioinertness, bioactivity, toxicity, carcinogenicity, thrombogenicity, immunogenicity. These aspects require knowledge of biology and blochemistry besides materlafs science. We will not be able to treat them all here, but in saveral places we will mention material properties that are important in biomedical engineering.

1.2.6 Bulk vs. surface

There are large differences in composition, and therefore also in properties, between the surface of a materialand its bulk (the "interior" or average). For biomaterials, this is especially important, since the interaction between material and human body occurs at the surface of the materiaL Therefore this is important for a lot of the biorrièdical aspects mentioned in· the previous·sectiori. · '·

2 Mechanica I

properties

In order to be a bie to speak a bout properties as strengthand stiffness, we should first look at a bit of mechanics. We shall see that especially the term "strength" is very vague, and should always be replaced by something more specific.

We will only discuss mechanica! behaviour of deformation in one direction.

Science is a long history of learning how not to fool ourselves.

Richard Feynman (1918-1988)

8 Materlafs Science for Biomedical Engineering

2.1 Elastic behaviour

Elastic deformation occurs when a force deforms a material, but after the force is removed, the original shape returns: it is reversible.

2.1.1 Deflnitions of stress and strain

Deformation of a material can occur in various ways, depending on the direction of the force applied to it. As can be seen in Figure 2.1, farces on a material (F [N]) can lead to various types of deformation.

A B c D

Figure 2.1. Different orientations of the force on a material can lead to tension or elangation (A), compression (B), bending (C) or shearing (D).

However, the force depends on the size of the material objects we are testing (thinner test bar: less force necessary). Therefore we always use stress, which is force per unit of area. Fortension and compression, the area in question is perpendicular to the force, and we use:

F cr [Pa]

Ao

Ao is the area of the original cross section of the tested materiaL In Figure 2.2 we see a typical elangation due to a tension stress. The cross section may have a different area after the elangation (A), but we willlook at that effect later in detail.

In order to describe what is happening to the dimensions of the material when a force is applied, we look first at the length change during tension or compression:

t.L L L0 [m]

2 Mechanica/ properties

However, this 11L is dependent on the length of the test bar, so we use the dimensionless quantity strain:

B 11L [-] Lo

Another dimensionless quantity that is aften used is the elongation factor (or stretch):

L A.=-=1+e

Lo

~I

F F

-l l F

F

Figure 2.2. Effect of a force in tension.

9

Forshearing (see Figure 2.3), the area we use for Ao is parallel to the force, and we write for the shear stress:

F • =~[Pa]

Ao

----F F AL -

F F

Figure 2.3. Effect of a shear force.


For shearing forces, we calculate the change in dimensions via the deformation angle:

y "' tan y = LlL (for small deformations, and y in rad) La

Now, La is perpendicular to LlL

Interme.zzo A special type of deformation comes from hydrastatic pressure (p [Pa]): the force is now the same trom all directions. For hydrastatic pressure, we look at the change in volume:

Ey

2.1. 2 Etostic de tormotion

Elastic deformation occurs when the material returns to its original shape after the force is released, like a spring: the deformation is reversible. This means that we are only stretching or campressing the chemica! bonds in the material, without breaking them. For elastic materials, Hooke's law holds:

cr EE for tension and compression 1: Gy for shearing

Here, Eis the Young's modulus ([Pa]), or modulus of elasticity, and G is the shear modulus ([Pa]). Both are a measure of the stiffness of a material: the resistance to elastic deformation. Both E and G are material properties, and can be found in tables (see e.g. Appendix E).

We will see later that the stiffness is not a[ways the same for tension and compression. For several compounds, tension is not always the most practical way to test stiffness, so bending (flexural stiffness) is often used.

lnterme.zzo In fact, stiffness is a proparty of a piece of material: when a force F gives a length change l!.L, the stiffness is Fll!.L [N/m]. The Young's modulus is a material property, and is related to the stiffness as E = stiffnessxLo/Aa.

I nterme.zzo Gompression due to hydrostatic pressure gives a bulk modulus K:

LlV p=-K-

Vo

"• '·

2 Mechanica/ properties 11

2.1. 3 Poisson ratio

The Poisson ratio (v) is used tomaasure the change in cross-sectional area during deformation, since there is also strain (usually negative: compression) in the x- and ydirection when we stretch a material in the z-direction. For isotropie materials, we can write:

Ex Ey V=--=--

Ez Ez

(Fora cylinder, the change in the diameter is used: Ex= Ey =Ad/do.)

L,i -

Figure 2.4. Change of all dimensions after deformation due to a tensi/e stress.

The volume of the material after deformation can be calculated now, since we knowhow the dimensions change (see Figure 2.4):

Lx = Lx,O ( 1 + Ex )

Ly = Ly,0(1 + Ey)

Lz = Lz,O ( 1 + E z )

V= LxLyLz = Lx,OLy,oLz,o(1 +Ex )(1 + Ey )(1 + Ez)

Using the Poi~~.on.raoo..t-Rls-becomes: ....__._..~··- .... ~--·"" .. -~---··

V =V0 (1-VEz)2 (1+Ez)=V0 {1+(1-2v)Ez +v(v-2)E~ +v2E~} p. " I"- ~ r I -· \I U ) 1

For small values of Ez (as is usually the case for elastic behaviour) this gives approximately:

AV = V-V0 =(1_ 2v)E Vo Vo z


Th is means that for v = 0.5, the volume of the material will nat change during tension or compression, which is aften assumed to make things easy. This is typical for rubbers: they can easily change shape.

lf v = 0, then ~V is large, since you can stretch the material while its cross-sectio na I area stays constant. This usually means that the bands between atoms are very strong, so you cannot stretch very far.

For most materia Is v = 0.2-0.3: an intermediate situation where ~VIVo is ca. 0.4Ez to 0.6Ez. Since most materials cannot be stretched elastically very far, this is nat a large effect.

Example: Aluminium alloys have v = 0.3, and can be stretched elastically only up to ~:

= 0.005. This means: ~VIVo= 0.002 at most (0.2%).

The following relation holds for the various measures for stiffness:

G=-E-2+2v

Th is means that for the usu al assumption of~ V= 0 (v = 0.5), we get: E = 3G.

2.2.4 flostic energy

When we plot stress versus strain, elastic deformation will be a straight line (cr = E~:). The area under this line is the energy that is necessary to deform 1 m3 of the material up to strain ~::

Since the deformation is reversible, this energy is stared in the materiallike in a spring: it is released when we remave the force. The maximum elastic energy that can be stared in a material is called its resilience. This maximum is reached upon fracture or yield (see next section).


2.2 Tensile test: plastic behaviour, fracture

Some materials, like ceramics, break when they reach the limit of elastic deformation. Most metals and polymers can be deformed further, but this deformation will not go away when the force is removed: it is a permanent, plastic deformation. We are then breaking chemical bonds in the material, and forming them again in a different position. To do this costs more energy, and is not reversible. However, it does not break the materiaL

The change from the elastic region to the plastic region is called yield. In the stress-strain curve there is a stress where yield occurs, which is called the yield strengthor yield stress (oy), and an associated yield strain (ev). What is the ditterenee here between stress and strength? The stress is sernething we apply from the outside, but seen as a material proparty it is called strength.

The stress-strain curves for the different types of materials are quite different: in Figure 2.5 we can see how ceramics show brittie fracture after elastic deformation (A); some metals and polymers will show yield and then fracture (B); other metals and polymers can be plastically deformed very far after yielding (C); and rubbers can be stretched elastically, but the relation between stress and strain is not linear (D).

/. yield

~ D

Figure 2.5. Different types of behaviour during tension: e/astic (A), plastic (B, C), and rubber-type e/astic (D).

Another important ditterenee can be seen when you cbserve the behaviour of materials during first stretching and then relaxing again. As depicted in Figure 2.6, matenals can be completely or linearly elastic (A) like ceramics; e/astic-plastic, where a part of the deformation is reversible (B), like most metals; completely plastic (C) as polymers can be; and visco-elastic, where the deformation is completely reversible, but the curves are not the same for tension and release (D). During this latter process energy is lost. In general this last type of behaviour is called hysteresis.

14 Matenals Science for Biomedical Engineering

A 8 (j (j

c D (j (j

Figure 2.6. Different types of behaviour during a tensile test where force is first applied and then removed: completely reversible (A), partly irreversible (B),

completely irreversib/e, and reversib/e with lossof energy (D).

When a metal or polymer undergoes a tensile test, it will first show elastic stretching, and then yield and plastic deformation. lt does nat start flowing like a liquid: the stress still rises, so the material continuestoresist deformation. This is called strain hardening or work hardening, and the cause for it will be explained in Chapter 8.

However, at a certain strain, a maximum stress is reached: the (ultimata) tensile strength (crr, see Figure 2.7). Usually, the material can be stretched a bit further, until it breaks, at the breaking strengthand the maximum strain (smax). lt may seem strange that the material breaks at a lower stress than the tensile stress, but this is an artefact of our definitions, as will be explained in the next section.

The lengthof the (two combined pieces of the) material after breaking is used to calculate the ductility:

Lfinal -La (aften in %, so then multiply with 1 00) La

Also, the reduction in cross-sectional area is used to indicate ductility:

Th is is easily related to the change in lengthwhen ~V= 0.


(j

breaking strength \

O'y

Sy

Figure 2.7. A tensile testfora material showing yield.

I ntenne..z;zo Usually the transition from elastic to plastic behaviour is not very clear. Often, the yield point is then defined as the 0.2% offset yield: a line is drawn parallel to the linear part of the curve, starting at e = 0.002. The intercept with the curve is then defined as the yield point.

CJy ---------------7

/ I

0.002 e

15

Since there may besome elastic recovery after breaking (elastic-plastic behaviour), the ductility can be smaller than expected from Emax· In Figure 2.8, cry,1 is the yield stress at the first elongation. At higher stresses there is plastic deformation, and after release of the stress elastic strain recovery is seen. The double-headed arrow

16 Materlafs Science for Biomedica/ Engineering

indicates the recovery, but the material stays permanently elongated. crv.2 is the higher yield stress when the material is stretched a second time.

cr

cr ·----------------Y,2

Figure 2.8. Elastic recovery may not be complete, giving a higher yield stress in the next tensile test.

The total area under the strass-strain curve is again energy per m3 (compare with the elastic energy): it is the total energy necessary for breaking the materiaL This is usually called the toughness of the materiaL Of course, if the ductility is high, so will be the toughness.

There are some ramarks we must make about fracture, because there are two quite different kinds of fracture (see Figure 2.9). Caramies will usually show brittie fracture: this is a very fast process at the end of the elastic region. Metals usually show ductlle fracture, as seen in the tensile test: a slowar process that involves necking at aT (see section 2.3) and plastic deformation. However, metals can also undergo the faster brittie fracture at any point of the tensile test, especially at lower temperatures. Which kind of fracture occurs first for metals will be investigated in Chapter 5.

(-..L.--J ------~s

(.......__) _> < _ ___,) Figure 2.9. Brittie fracture (top) vs. ductile fracture (bottom).


Where in the stress-strain curve can we find the various mechanica! properties? This is demonstraled in Figure 2.10.

Stiff materia Is have a high slope in the elastic region (high E), while flexible or compliant or pliant materials have a low E.

Brittie materials have a large resistance toplastic deformation (high crv). while ductile materialscan be deformed easily (low crv).

Strong matenals have a high tensile strength (crr}, while weak matenals have a low crr. However, some people call a material weak when it shows yield at low stresses (low crv}. Of course, when the material doesnotshow yield (such as ceramics), the strength is determined by the end of theelastic region.

A tough material can absorb a lot of energy before breaking (large area under the curve): it usually can be deformed a lot (high ~>max). lf the area is small, the material is brittle.

cr st rong

brittie

brittie ..,. .. _!-----+~ tough

stift

Figure 2.10. The various machanical properties in the stress-strain curve.

A proparty that is often mentioned, and that can be very easily measured, is the hardnessof a materiaL lt represents the resistance to local plastic deformation, such as scratching or indentation. lt is defined purely empirically, and can not be easily related to the properties above. For metals, it is related to the tensile strength, and also to the yield stress. For non-metals, it is related to the lattice energy: the strength of the chemica! bonds. Forthese materials, it is also related to the yield strength in compression. We will not discuss it in more detail.


2.3 True stress and straln

fn the stress-stmin curve of metats, we saw a strange behaviour tor ductite fracture: there is a maximum stress (crT). but the material breaks at a lower stress and a higher strain. How can this happen?

The cause is our definition of stress, which is related to the original crosssectionat area. This is called the engineering stress:

F cr=-

Ao

Th is is nat the actual stress in the material, because the area of the cross-section will usually decrease during the tensite test (or increase during compression). We therefore must look at the true stress:

F O"t =-

A

lf we now assume that the volume does notchange during tension, then we can say: A/Ao = Lo/L, and this leads to:

The relation between true and engineering stress is then:

For small values of e, crt and a are approximately equal. ---="~-'I -r However, at the maximum engineering stress, the material showslneckinJJ; at

one location, the area A suddenly beoomes smaller very rapidly (see1"igUl'Ef2.11 ). The true stress (cr1) still increases after this point, but since the area (A) decreasas faster than the true stress increases, the engineering stress (cr) goes down:

For more details, see Appendix B.


(j

------------------------~--::.::o--=---~

!

By Bmax S

Flgure 2.11. Necking occurs at the tensile stress, leading quickly to ductile fracture.

In Figure 2.12 we see the engineering stress (solid line) and the true stress (dashed line) for a tensile test:

stress

strain

Figure 2.12. True stress keeps on increasing (dashed line), even when the engineering stress goes through a maximum (solid fine).


In order to describe the behaviour of the true stress befare the necking, it is a lso necessary to use another maasure for the strain, th7'~ stra!!9

L dL ( L J s1 = J- In- =ln(1+s) 4 L Lo

At I(

While the engineering strain e r~ns from -1 to +oo, et has a more symmetrical range tram -oo to +oo. In Figure 2.13, s1 is plotled as a tunetion of e. For small values of e, e and St are approximately equal.

-1

Figure 2.13. Relation between true strain and engineering strain.

We can now describe what happens during the tensile test. 1. In thee/astic region: cr = Ee, and also: cr1 = Es~o until yield: e =et= ey. 2. In the plastic region it is found that there is a simple, empirica! relation between true stress and true strain: . -~ .•. A •.•. "'~. • \j IJYOv' c: '"~' .>

' ' ..!. ..... ' ' 'y; N Y ~ r<.'""""·~~:( \.;, 'J vV <: •:. ~"""'"- '/> .}

cr, Kp,et "<) V# . "' . I, "' . , ,,)_ 6 \.Al'~ ~'C {_ V .-- V

Kp1 and N are material properties; N is called the strain hardening exponent. lt can now be shown {see Appendix B) that necking occurs when the true strain equals the exponent N:

St = N 1,w { "';> p;t;,y;,:-3. After the necking the true stress still increases, but this is very difficult to describe, since the necking will quickly lead to ductile fracture.

Example: A material yields at an engineering strain sy = 0.002, has a Young's modulus E = 1 GPa, and N = 0. 1. Necking will then occur at st= 0.1, are= 0.11. In the elastic region we have cr = Ee, and cr1 = Es1• At the yield point we must have: crv = Esv = 0.002 GPa. crt and e, are approximately the same as cr en s at this point.

/

2 Mechanica/ properties 21 ~ r ' 1 I · '•· •I • L

AAAA(C" ~C,-:;.·!J)· Lc~,., .. (;••C ;:l(V)•t?"" ':J .. XAr.ClL. 'fiV' L·f ifll(lf'A-' ,. Ó (.\ ' 't C

• vw After the yield point we have":lcrt = Kp1Er until necking. After necking, at will increase until fracture, so the cr1,s1-curve looks like Figure 2.14.

0.5

0.1 0.2 0.3 0.4 0.5

Et

Figure 2.14. Relation between true stress and true strain in the plastic region.

For ~he _engineering stress in the plastic region, it can be w~en that (still assuming

.!lV- 0). ~ t. v-:. o .>, <ï_aePc\ t ~ ~--,-o l\-IC()

a=K I {ln(1+s)}N p 1 + E

Therefore, the cr,s-curve shows a maximum at s = 0.11 (necking), as shown in Figure 2.15.

0.5

0.1 0.2 0.3 0.4 E

Figure 2. 15. Maximum in the engineering stress-engineering strain curve as an artefact of the definitions.


2.4 Overview of material properties

In Figures 2.16 and 2.17, a graphicet overview is given of some typical ranges for the mechanica! properties of saveral types of materials. We will see in Chapter 5 why normalized yield stress and normalized toughness are used.

10-3 1Q-2 10-1 10° 101 102 103 10-7 10-6 10-5 10-4 10-3 10-2 10-1

caramies

polymers

polymers

biomaterials metals

.. limit limit

Figure 2.16. Left: Young's modulus (E [GPa]); right: normalized yield stress (av/E).

Fora lot of applications, materials have to be heated-for processing, shaping, or sterilizing-and the melting temperature (in kelvin) can be very important.

10-1 10° 101 102 103 104 105

...____... ceramics

.. polymers

metals

limit

0 1000 2000 3000 4000

..___.. polymers

caramies

metals

Figure 2.17. Left: norma/ized toughness; right: melting temperafure [Kj.

limit

There are various other graphical methods to quickly campare different materials, where e.g. stiffness (E) and weight (p) can be compared, or strength (crr) and weight (p). Also, the three propertiescan be combined in a diagram of reduced strength (crr/p) vs. reduced stiffness (E/p). Thesediagramscan be found in various books, and are very useful for selecting materials.


Problems

2.1.a. What is the final length of a 2 m long cylindrical bar of copper, 0.01 m in diameter, stressed by a 5 kN force?

23

b. lf a steel bar of the same diameter has the same force applied to it, how long must it be initially to extend the same amount as the copper bar in a.?

2.2. A specimen of Al with a rectangular cross section (1 0 mm x 12.7 mm) is pulled in lension with 35.5 kN, producing only elastic deformation. Calculate the rasuiting strain.

2.3. A cylindrical Ti implant with diameter 19 mm and length 200 mm is deformed elastically in lension with a force of 48.8 kN. a. Calculate the elongation. b. Calculate the change in diameter.

2.4. A steelpipeis hanging vertically down from a support. The pipe has an outer diameter of 7.5 cm, an inner diameter of 5 cm, a density of 7.90 Mg/m3

, and a yield strength of 900 MPa. Assume a safety factor of 2. a. What is the maximum length of the pipe before yield occurs? b. What is the maximum elastic strain in the pipe, and at what location along the

pipe does it occur? c. At what location along the pipe does the elastic strain have a minimum value?

2.5. The engineering stress-strain curve tor a given implant metal can be described by three straight lines, with cr in MPa. Elastic: cr = 138000s for 0 < e < 0.0025 Plastic: cr = 327 + 7075s for 0.0025 < s < 0.1 and: cr = 1725- 6900s for 0.1 < e < 0.15 a. What is the value of Young's modulus? b. What is the value of the yield stress? c. What is the value of the ultimata tensile stress? d. Calculate a value for the resilience. e. Calculate a value for the toughness. f. What is the percentage elongation?


2.6. A specimen of ductile cast iron with a rectangular cross section (4.8 mm x 15.9 mm) is deformed in tension. The results are given below.

Load/N

0 4310 8610

12920 16540 18300 20170 22900 25070 26800 28640 30240 31100 31280 30820 29180 27190 24140 18970

fracture

Length/mm

75.000 75.025 75.050 75.075 75.113 75.150 75.225 75.375 75.525 75.750 76.500 78.000 79.500 81.000 82.500 84.000 85.500 87.000 88.725

Delermine the following properties: a. The modulus of elasticity. b. The tensile strength. c. The resilience. d. The ductility in percent elongation.


2.7. A tensile bar has a circular cross section of area 0.2 cm2. The applied loads

and corresponding lengths recorded in a tensile test are:

Load/N 0

6000 12000 15000 18000 21000 24000 25000 22000

Length/cm 2.0000 2.0017 2.0034 2.0042 2.0088 2.043 2.30 2.55 3.00

fracture

After fracture the bar diameter was 0.37 cm. Delermine the following: a. Young's modulus. b. Ultimata tensile stress. c. Percentage reduction of area. d. True fracture stress.

2.8. For a hardened steel alloy the true stress-strain behaviour is: crt = 1310st0·

3 [MPa] a. What is the true strain at necking? b. What is the engineering strain at necking? c. What are the true and engineering stress levels at necking?

2.9. A tensile test is performed on a metal specimen, and it is found that a true plasticstrain of 0.20 is produced when a true stress of 575 MPa is applied; for this metal, the value of Kp1 is 860 MPa. a. Calculate the true strain that results from a true stress of 600 MPa. b. Calculate for both true strains the corresponding engineering strains and

stresses.

2.1 0. Wire used by orthodontists to straighten teeth should ideally have a low modulus of elasticity and a high yield stress. Why?

25

2.11.* A cylindrical stainless steel wire must be implanted. The steel has a Young's modulus of 193 GPa, and yields at s = 0.0015. a. How large must the diameter of the wire be to show noplastic deformation at a

laad of 900 N? lf there is yield at higher loads, we want to prevent necking, since this willlead to fracture and damage for the surrounding tissue. For the plastic deformation of this steel it is known that: Kp1 = 1275 MPa and N = 0.45. b. Calculate the force where necking occurs for this wire. Clearly indicate all

assumptions.

26 Materlafs Science tor Biomedical Engineering

3 Bonds between

atoms:

Microscopie structure and elastic modulus

There are several types of bonds that can keep materials tagether (see Chapter 1 ), but most of them can be modelled in a simple way as a combination of attraction and repulsion terms.

Elastic deformation occurs through the stretching or compression of bonds between atoms, so in principle we can calculate the Young's modulus E of a material if we knowhow the bonds behave. Th is is the only mechanica! property we can calculate from the mieraseale structure.

Live as if you were to die tomorrow; leam as if you were to live forever.

Mohandas Karamchand "Mahatma" Ghandi (1869-1948)


3.1 Bond energy

All bonds have an attmction between atoms due the chemica! bonding, which lowers the energy, and a repulsion between atoms due to the repulsion between the positively charged nuclei, which prevents atoms from coming too close.

A formula that is used a lot to describe the energy in such a case is:

A B U=--+- [J]

Rm Rn

Here, A and m reprasent the attractive term, and B and n reprasent the repulsion. lt is mostly found that n = 12. For van der Waals forces m = 6 is often used, and for torces between dipoles m = 3. The interatomie distance R is the sum of the two atomie (or ionic) radii. The relation between energy and distance can beseen in Figure 3.1.

Repulsive energy (8/R'l)

/ Energy (U)

Interatomie distance (R)

\ Attractive energy (-Al Rm)

Figure 3. 1. The energy in a bond between two atoms or ions is built up from an attractive and a repulsive term.

For electrostalie forces, which are important in salts, the attraction term is given by Coulomb's law for two charges (q1 and q2), separated by a distance R:

3 Bonds between atoms 29

So, for the attraction energy of ions in salts we have m = 1, and, by writing q1 and q2 as multiples of the unit charge e (q1 = z+e for the cation and q2 = z_e for the anion), we get:

Therefore the total energy between ions in salts can be written as:

Two atoms form a stabie bond at the distance where the energy U is minimal, so when:

dU =O dR

Using the general formula for U, this leads to an interatomie distance:

(

nB )1/(n-m) R0 = - [m]

mA

With this relationship, one can later on always eliminale B from all formulas via:

mARn-m B=- o

n

which is often useful. The energy at the equilibrium distance R0 is now defined as:

Uo = U(Ro)

For electrostalie bonds in salts, where m = 1, eliminatien of B leads to a simple formula for the equilibrium energy for two atoms (check this!):

A ( 1' -- 1--j [J] R0 n


Inter11ZUZo In real materials, there are of course more than two ions present. Some ions are oppositely charged, and others have the same charge. Also, not all ions are at the same distance from each other. lf you average the effect of all other ions surrounding one ion, the formula tor the attractive energy is not that different trom the one we derived:

The only difference is the so-called Madelung constant A, which is different for each salt. E.g. for NaCI: A= 1.7476, and for CsCI: A= 1.7627.

This constant is obtained by adding the electrastatic contributions trom ions at different distances: e.g., the ciosest 6 are attractive, the next 12 are repulsive, the next 8 are attractive again ...

A= 6 -12x1/v'2 + 8x1/v'3

To understand these numbers, you should take a look at Chapter 4 on crystal structures.

The repulsive force is then introduced in the so-called Born-Landé equation exactly as we have done:

The n in this equation is also dependent on the ions involved. lt is the average of the values tor the ions from the list:

n ion

s w, u+ 7 F-, 0 2-, Na+, Mg2+ 9 er, 52-, K+, Ca2+, Cu+ 10 Br-, Rb+, S~+. Ag+ 12 r, Cs+, Ba2+, Au+

As you can see, n is related to the size of the ions.

The simpte theory tor 2 ionsis therefore quite good.


3.2 Porces on bonds

The force between the atoms can be easily calculated from the energy curve in the usual way:

In Figure 3.2, we can see the force between the atoms. A positive force here means repulsion, so the atoms want to move apart, and a negative force means attraction, so the atoms want to move closer together. The force is exactly zero at the equilibrium distance, as it should be.

Interatomie distance (R)

Figure 3.2. The dependenee of interatomie torces on the distance.

Usually, we are interestad in the force we need to deform a material from the outside, so we use the formula with an inverted sign:

Now we have the usual behaviour as seen in Chapter 2: a negative force is needed to compress the material (atoms closer together), and a positive force is necessary to stretch the material (atoms further apart). This is illustrated in Figure 3.3.


Force (F) Interatomie distance (R)

Figure 3.3. The force needed extemally to change the distance between the atoms.

For elastic behaviour, we are interestad what the resistance is against small changes in the bond length between the atoms. Only tor very small detormations, we can approximate this as the behaviour of a spring by drawing a straight line:

The spring constant ks can be calculated trom the slope of the F,r-curve at R = R0:

k =(dF J =-m(m+1)A+n(n+1)B s dR Rm+2 Rn+2

R~R0 0 0

Th is looks quite complicated, but with the relation we saw earlier for R0, we can write B = (mAin)Ron-m, and then we get:

k = m(n m)A s Rg'+2

We can now relate this spring constant ks tor two atoms to the Young's modulus E of the material by realizing that the strain here is:

R-R0 E=--Ro

Since the atoms have distance R0, this is also roughly their diameter, and they therefore use an area of about Ro2

, the stress is:


With F = ks(R- Ro), we then get:

This very rough approximation gives reasanabie predictions for the Young's modulus. Of course, we can perfarm more realistic calculations when we know more about the bands.

ExampJelE~~§<:~It$_111{1::Lknow.that.m.=.1-', and that A= -2.3·10-28z+z _ for ionsof charges i. and z -· lf we then know the values of n and R0, we can calculate E. For NaCI, it is known that Ro = 0.282 nm, and that n = 9.4. Also, the ions have charges +1 and -1. This leads to the prediction: E = 305 GPa. Experimentally it is found that: E = 49 GPa, so we are only a factor 6 off.

3.3 General characterlstlcs

Meta/s: The metallic bond is quite strong (cohesive energy: 100-800 kJ/mol), and is isotropic: it has the same strength in all directions ("electron sea"). We will show in Chapter 5 that this leads to the easy deformation of metals. The predieled Young's moduli are 60-300 GPa, and this fits quite well with experimental values.

Ceramics: The bands are very strong (ionic: 600-1500 kJ/mol, or covalent: 300-700 kJ/mol), but anisotropic: they depend on the location of the nearby atoms. We will see in Chapter 5 that this leads to the brittleness of these materials. Th is holds especially for covalent networks, where the bands are located exactly between pairs of atoms. In ionic bands, the bands are directed between the + and - charges on the cations and anions. For covalent bands a Young's modulus of 200-1000 GPa is predicted, and for ionic bands: 32-96 GPa. These values fit quite well with experimental values, although for ionic ceramics they are on the low side.

Po/ymers: Bonds between macromolecules are quite weak (van der Waals, H-bonds: 10-50 kJ/mol). The predieled values for the Young's moduli are 8-12 GPa (H-bonds} and 2-4 GPa (van der Waals), but many polymers have much lower moduli. This is due to the fact that most polymers are in fact not completely solid, but partly molten ("liquids in slow motion"). We will discuss this in more detail in Chapters 6 and 7.


Problems

3.1. A common form for the potentiat energy of interaction between stoms is glven by U= -A!R 6 + B/R12

, where A and Bare constants. a. Derive an expression for the equilibrium distance between the atoms in terms of A

and B. b. lf Ro = 0.25 nm, what is the ratio of B toA? c. Derive an expression for the energy at the equilibrium separation distance in

termsof A.

3.2. The ionic solids LiF and NaBr have the same structure as Na Cl. For LiF, n = 5.9 and Ro = 0.201 nm, whereas for NaBr, n = 9.5 and R0 = 0.298 nm. a. Which of these materials is expected to have the highest modulus of elasticity? b. Calculate the molar energy of ionic interactions for both materials. LJ l ~ o) 6

3.3.a. Opposite sides of a rocksalt (NaCI, with n = 9.4) crystal are pulled to extend the distance between neighbouring ions from Ro = 0.2820 nm to R = 0.2821 nm. Similarly, during compression, ions are squeezed to a distance of 0.2819 nm. Campare the value of the tensile (or extension) force with that of the compressive force. Are they the same?

b. Repeat the calculation if the final distances are 0.2920 nm for extension and 0.2720 nm in compression.

3.4. The potential energy U between two ionsis somatimes represented by:

U c D -Rip --+ e R

a. Derive an expression for the bond energy Uo in terms of the equilibrium separation R0 and the constants D and p.

b. Derive an expression for U0 in terms of R0 , C, and p.

3.5.* We can treat zirconia (Zr02) as an ionic ceramic with a rocksalt structure, consisting of Zr4+ and 0 2

- ions. Calculate the exponent n of the repulsion term in the interaction energy between the ions from the Young's modulus.

nfl" u --I ::. o) o 1? ~ /""J'f'f'.

)\.\9-c-\ - OJ \ j /_ ('1\fl(\

4 Inorganic

Materials 1:

Structu re of crystalline metals and ceramics, and glasses

Metals and ceramics are among the most important "strong" construction materials, in contrast to the "soft" polymers. They are used a lot as engineering materials, but also as biomaterials.

lnorganic materials are mostly crystalline, so in this Chapter we will first focus on the structure of crystalline materials, and we will take a short look at non-crystalline glasses.

In the next Chapter we will see what the implications of these structures are for the mechanica! properties of inorganic materia Is.

It is not by prayer and humility that you cause things to go as you wish, but by acquiring knowledge of naturallaws.

Bertrand Russell (1872-1970)


4.1 lntroducflon

In inorganic materials, the bands are ionic, covalent or metallic, and the structures are mostly crystalline. Such structures are highly symmetrie, and can bedescribed by repetition of a unit cell. Deformation of these materials is mainly determined by the bands between the atoms and by imperfectlans in the crystal structure. When a crystalline substance loses its order, we call it a glass.

4.1.1 Bloceramies

In the human body we do not find metals as natura! materials, and almast nopure ceramics. Most caramies are present as part of a composita material, such as bone or tooth (see Chapter 9). In these composites, a large control is seen in the way the ceramics are formed, guided by organic polymers. Because biologica! composites are formed by growth, the formation of the crystals can be tuned exactly in time and structure (biomineralization ).

The minerals most aften found as bloceramies are calcium carbonale (CaC03),

hydroxyapatite (Cas(P04h(OH)), and silica (Si02). Only hydroxyapatite is found in the human body. C.aC03 normally farms~ calcite crystals, but a number of organisms can change the crystal growth completely fortheir own purposes. Also, Si02 is used for many beautiful structures, such as in diatoms.

I nterrn.ezzo Biomineralization is also called "molecular tectonics": the building of structures with molecules. This process has a number of stages: - The organic molecules organize themselves, e.g. as polymer networks or vesicles. - There is recognition between charged groups on the organic surface and the inorganic ions. Th is is determined by the location and nature of the charged organic groups. - The morphology (shape) of the growing crystal is now directed: some planes of the crystal can grow, and others not, by adsorption of organic molecules. - Various small crystals are organized in larger structures. For more information: S. Mann, Blomimetic materials chemistry.

4.1.2 lnorganic blomater/als

Metals that are used a lot as blomaterial are: titanium, stainless steel (Fe with C, Cr, Ni, Mo, Cu), cobaiUchromium alloys (e.g. Vitallium: Co2Cr with Ni and Fe), and gold. They are used a lot as replacements for joints, plates and screws in bone, and root implants in teeth.

Metals are aften strong and tough, but they have a high density and can corrode in a biologica I environment. A typical biomedical application is amalgam for fillings: an alloy of 60% Ag, 29% Sn, 6% Cu, 2% Zn and 3% Hg is dissolved in mercury. Then, the Ag3Sn starts to react with Hg, and Sn1Hg and Ag2HQ3 are formed. The final solid has the composition: 45-55% Hg, 35-45% Ag, and 15% Sn.

Caramies that are used a lot as blomaterial are alumina (polycrystalline AI203; hip prosthesis, tooth components), zirconia (Zr02, orthopaedic and dental applications), hydroxyapatite (bone and tooth replacement) and carbon (heart valves, ligaments,

4 lnorganic Materials I

dentistry). Ceramics are aften inert, biocompatible and strong in compression, but they are also brittie and hard to shape.

Glasses are employed in orthopaedic applications and as bone replacements. Also the glass-ceramics are used, such as Macor, which has mica crystals mixed with a glass consisting of Si02, 820 3, AI203, MgO, K20 and F-. Then there are the bioactive glasses, which can integrate with bone, such as Bioglass (CaO, P20 5 ,

Na20. and Si02). Drugs can also somelimes be oftered better as glasses than as crystalline

powders, because they will then dissolve more quickly. This happens because the glassy state is less stabie than the crystalline state (see section 4.6).

Interme.zzo "Bioceramics": Gems, wom on the human body - Corundum (Ab03): colourless crystals. With Cr added: red rubies. With Fe and Ti added: blue sapphires. - Quartz (Si02): colourless crystals. With FeO added: purple amethysts. With MgO and FeO added: brown-red layers in agate. Amorphous Si02: milky white opal. - Beryl (BesAizSi601s): hexagonal crystals. With Cr added: green emeralds. - Diamond {C): cubic crystals.

37

The well-known vibrations given off by crystals and their effect on human health are not treated here. © Also, the healing magnatie fieldsof metals are ignored. ©©


4.2 CrystallaHices

When we want to describe the structure of a crystalline substance, we must specity the unit cell, and the atoms in it. The unit cell is the repeating unit in the pattem of atoms in the crystal. For simplicity, we will first look at a repeating pattem in two dimensions (like wallpaper). As can be seen in Figure 4.1, we can find saveral unit cells in this lattice, but we usually take the smallest possible .

• • • • • •

• • •

n ·----------·

•

• Figure 4.1. A two-dimensionallattice with several possible unit ce/Is.

Mathematicians have been studying the symmetries of such pattems and their unit cells, and have found that there are just a few types of unit cells possible (see Figure 4.2). Usually, the following narnes are used: square (highest symmetry), rectangular, 120° rhombus (or hexagonal), rhombus (or centred tetragonal), and parallelogram (no symmetry).

D D

Figure 4.2. The unit ce/Is in two dimensions that are unique.

4 lnorganic Materlafs I 39

These unit cells can be classified according to the length of their sides (a, b), the socalled lattice parameters, and the angle between the sides (y), as shown in Table 4.1.

Tab/e 4.1. The different unit ce/Is in two dimensions.

Lattice type 'Y = goo 'Y = 120° 'Y * goo a=b square hexagonal centred rectangular 8*-b rectangular - parallelogram

An important trick for working with crystal lattices is counting the number of lattice points per unit cell: since every point on a corner belongs to 4 unit cells, it belongs for only Y4 to one cell. lf there are four lattice points on the four corners, then only 4 x Y4 = 1 point belongs to one cell.

Sarnething similar can be done in three dimensions, by placing these planar lattices on top of each other. Then the following crystal systems are found: Figure 4.3 shows the cubic (most symmetry), tetragonal, hexagonal, and rhombohedral (or trigonal, somatimes combined with hexagonal) system, and Figure 4.4 shows the orthorhombic, monoclinic, and triclinic (least symmetry) system.

D-A ~ ~

D-A

,,//

' ' ' ' ' ' ' ' ' ' ' ' ' l

·-

Figure 4.3. The three-dimensional unit ce/Is with the highest symmetry. The eye symbol shows the result of a top or side view.

40

D-A ' ' ' ' ' ' ' ' '

/~-------l/

Materials Science tor Biomedical Engineering

/z---~~~v-A ó-----------6

b

Figure 4.4. The three-dimensional unit ce/Is with less symmetry.

These crystal systems can again be classified by the length of the sides (the lattice parameters a, b, c) and the angles between the sides (a, ~. y), as shown in Table 4.2.

Tab/e 4.2. C/assification of the three-dimensiona/ unit ce/Is.

Crystal system a=b=c a=b*c 8*-b*C a=~= y = 90° cubic tetragonal orthorhom bic a = ~ = 90°, y = 120° - hexagonal -a=~= y * 90° - - monoclinic a= y = 90° * ~ rhombohedral - -a*~*Y - - triclinic

Because you get unit cells with different symmetries by placing additional lattice points on the side faces or in the centre of the unit cell, there are in total14 Bravais lattices (see Figure 4.5). We will be dealing only with the cubic and hexagonal structures, since they are most aften found for metals and ceramics.

4 lnorganic Materials I 41

c

I!!J c a a

a

Figure 4.5. The Bravais lattices, of which three are cubic (A), tour orthorhombic (B), two tetragonal (C), one hexagonal (0), one rhombohedral (E), two monoclinic (F),

and one triclinic (G).

A very important point to remember is that all patterns up to now consist not of atoms, but of mathematica/ points in space. Our task is now to attach one or more atoms to each point, in order to make a real crystal structure. Of course, we must attach exactly the same group of atoms in the same orientation to each point, si nee they are mathematically identical.


4.3 Cubic and hexagonal structures

The cubic crystal system has three lattices: - Simple cubic (SC), with 8 x Ye = 1 lattice point per unit cell - Body-centred cubic (BCC), with 8 x Ye + 1 = 2 points per cell - Face-centred cubic (FCC) with 8 x Ye + 6 x Y2 = 4 points per cell. Now we must look at real materials: one or more atoms must be attached to the lattice points.

4.3. 1 Cubic structures

SC: 1 atom per lattice point in an SC structure (see Figure 4.6) almost never occurs; polonium at ca. 1 0 oe is an example. Of course, the atoms should be touching along the edges of the cube (a = 2r = R); they have been drawn smaller to get a clearer picture (this is also done for the other structures). Each atom actually belongs to one unit cell for only 1/8th. Therefore each unit cell only contains 1 whole atom ..

Figure 4.6. The SC unit cel/ with 1 atom per lattice point.

More aften, 2 atoms per point is seen for SC, e.g. in caesium chloride (CsCI). This is the CsCI-structure (see Figure 4.7), with one atom on each lattice point, and a second (different) atom in the middle of the cube, and not on a lattice point. We still have 1 lattice point per unit cell, but now there are 2 atoms per cell. There is 1 atom of each kind per cell, so 2 atoms in total.

Figure 4.7. The SC unit cel/ with 2 different atoms per lattice point.

4 lnorganic Matenals I 43

For BCC structures, 1 atom per lattice point is seen almast always for metals (Fe, Na, K, Cr, Ta, V, W), as shown in Figure 4.8. These atoms then fill about 68% of the available space in the unit cell (actually, the fraction is 1t..J3/8; can you tigure out why?). This looks a lot like the CsCI-structure, but now all atoms are the same. For the situation with 1 atom per lattice point, we find that the spheres touch along the so-called body-diagonal of the cube (the line between opposite corners of the cube ).

a.V3 = 2R =4r

Figure 4.8. The BCC unit cel/ with 1 atom per lattice point.

In rare cases, we see BCC structures with more than 1 atom per lattice point (check the Table in Appendix G).

FCC structures with 1 atom per lattice point are also seen for many metals (Au, Al, Cu, Ag, Ni, pt, Fe between 910 and 1400 °C), as shown in Figure 4.9. Here, about 74% ofthe available space is filled (actually, it is 1t/(3..J2); why?). This is the maximum that can be obtained with spheres in a box: this is called a close-packed structure. In an FCC structure with 1 atom per lattice point, the atoms touch along the side diagonal of the cube.

=4r

Figure 4.9. The FCC unit cel/ with 1 atom per lattice point.

44 \

Materfals Science for Biomedica/ Engineerirtg

FCC structures with 2 atoms per lattice point are seen a lot for ceramics. With 2 identical atoms per lattice point, it is called the diamond cubic structure (e.g. Si, Ge, C as diamond), and we get tetrahadral structures (Figure 4.10, left).

When the 2 atoms are different, there are saveral possibilities. One of them is the zinc blende structure (e.g. GaAs, TiC, ZnS, TiN), which is basically the same as the diamond cubic structure with different atoms (Figure 4.1 0, right).

Figure 4.10. FCC unit ce/Is with 2 identical atoms per point (left), and with 2 different atoms per point (right).

Another possibility for an FCC structure with 2 different atoms per lattice point is the rocksalt structure (e.g. NaCI, MgO). As can beseen in Figure 4.11, this is different only in the orientation of the two-atom motif at each lattice point: it is now along the edge instead of along the body diagonal.

Figure 4.11. The rock sa/t unit cel/ is a/so an FCC ce/I with two different atoms per lattice point.

In the rock salt structure, the edge of the cube is related to the ionic radii as: a= 2r+ + 2c For ionic structures like CsCI, zinc blende, and rock salt, the relativa size of the ions also delermines the kind of packing (see Appendix L).


4.3.2 Hexagonal structures

Hexagonal structures are most aften displayed with a hexagonal unit cell (see Figure 4.5), which contains 12 x Ye + 2 x%= 3 lattice points. This hexagonal prism has volume: (\13)(3/2)a2c ....

Matenals with this structure are almast always seen with 2 atoms per lattice point (see Figure 4.12): these are hexagonal close-packed (HCP), and ca. 74% ofthe space is filled (the same as FCC). Examples are. Zn, Mg, and Ti.

Figure 4.12. The HCP unit cel/ with 2 atoms per point.


I ntenne.zzo A special case are the sHicates: (SI02)n. The basic unit Is an SI04 tetrahedron, which can be connected to other silicons at all 4 corners: we then get crystalline quartz. However, when notall corners are used we can get layers (e.g. mica, clay; C in Figure below) or strandsof silica (single chain, as A in Figure below, or double chain, as in 8), with cations to balance the charge.

A:

8:

AAA~

C:

Intenne.zzo Classica/ ceramics: C/ay -f bricklporcelain The common mineral feldspar (KAISi30a) can lose potassium in an acidic :environment to form HAISi30 8. Basic solutions then change this to kaolinite, AI4Si401o(OH)a, which precipitates as clay. • The structure of clay is an example of a layered silicate: there is a layer with 1Si20 5

2- as unit, and a layer with AI2(0H)/+ as unit. The interaction between the layers is weaker than the covalent bonds within the layers: clay can easily be sheared.

By heating clay to 700-900 oe it loses water, and brick or porcelain is formed: 2 AI203 + 4 Si02.

4 lnorganic Materlafs I

4.4 Polymorphism and polycrystallinity

Many substances can crystallize in a number of different crystalline structures, depending on temperature, solvent, pH, etc.: they are polymorphic.

47

Example: CaC03 (see Figure 4.13) normally crystallizes in the most stabie form, calcite (rhombohedral). This form is found in chalk and limestone. At higher temperatures, aragonite (orthorhombic) can crystallize. This form is found a lot in shells and coral. A form that is not stable, but changes into one of the others after a certain time (metastable) is valerite (hexagonal).

Figure 4.13. The three polymorphs of calcium carbonate.

Also important is the fact that only a tew matenals of reasanabie size are composed of a single crystal: this type of compound has properties that can be predieled from the crystal structure, and will show strong ditterences in properties between different directions (anisotropic).

Most materials that are used in practice are polycrystalline: they consist of lots of small crystals (grains) that stick together. The effect of this is that properties are averaged out over all directions (isotropic), and that the properties of the material are also determined by the places where the crystals stick logether (grain boundaries).

48 Materia Is Science for Biomedica/ Engineering

In Figure 4.14 we can see how a polycrystalline material is formed in genera!: crystals (squares) start growing inthemelt until they meet. The rasuiting liquid will solidify on one of the orystals, until everything is soltel in the form of grains with grain boundaries in between.

a b D

c d

Figure 4.14. Several stages of grain formation: (a) formation of first crystals, (b, c) growth of crysta/s until they meet, (d) final so/idification with grain boundaries.

4 lnorganic Materia/s I

4.5 Milier indices

In cubic crystal structures we want to be able to indicate positions, directions and planes, so we can teil in what way the structure changes upon deformation.

49

lt is important to remamber that each of these positions, directions and planes returns in all other unit cells. Also, the origin of the coordinates can be taken at any corner of the unit eelt, since they are all equivalent.

4.5.1 Positions

As can be seen in Figure 4.15, positions are indicated with x, y and z coordinates as usual, where the length of the edge of the cube is taken as the unit. The origin can be taken as any of the points of the cube, because they are all the same (the unit cell repeats in the lattice ).

Example: (0, %, -1) means 0 along the x-axis, a/2 along the y-axis, and -a along the z-axis.

z

z

! (%,%,%)

~----------+---------·--------· y (1 ,0:~!- __ ...-.... (0,0,0) (0, 1 ,0)

x (%,0,%)

x Figure 4.15. Se veraf examples of positions in a cubic unit cel/.

4.5.2 Dlrections

Directions are also indicated in the usual way as vectors: the ditterenee between two points. The notation is different than normal for vectors: Only integer numbers are used, and negative numbers are indicated with a bar over the number. The inlegers are then written between square brackets (see Figure 4.16).

Example: the vector from (0, 1 ,%) to (1,0,1) is normally (1,-1,%). Multiply by 2 to get integers: (2,-2,1 ). Now write this vector as: [2 2 1 ].


z [001]

r

[301]

[110]

-· [01 0] y

Figure 4.16. Several directionsin a cubic unit cel/ with their Milier indices.

Since a lot of different directions in a cube are actually the same due to symmetry, they are bundled tagether as a family: <hkl>.

Example 1: The notatien <100> indicates all similar directions along an axis: [100], [010], [001], [100], [010], [001]. Example 2: All body diagonals in a cube can be grouped tagether as <111 >.

Angles between veetors can be calculated via the so-called dot product ar sealar

product. When we have two veetors Ä = (a1,a2,a3) and B = (b1,b2,b3), we can calculate the dot product in two different ways:

(i) via the x-, y- and z-coordinates of both vectors: Ä · B = a1b1 + a2b2 + a3b3; (ii) via the lengths of the veetors and the angle between them:

Ä·B = IÄIIBI cos($), where IÄI = 'l/(a12+ ai+ a}) and IBI = 'l/(b12 +bi+ bl). The angle $ between two veetors [h1k1/1) and [h2k2/2) is then given by:


4.5.3 Plones

For planes the notation is a bit more complex: it is easier first to show how it works fora line. Every line can be given as:

y= ax+6

However, this can also be written as:

ax-y=-6

and then as:

a 1 --X+-y=1

6 6

With ~ =- 6/a and :B = 6, wethen get:

51

~ and :B now reprasent the intercepts of the line with the axes: x = 0 gives y = :B (y

axis intercept), and y = 0 gives x=~ (x-axis intercept). The same can now be done for a plane. Normally the equation for a plane is:

z = ax + 6y+ c

but this can also be written as:

x y z -+-+-=1 jf '13 c

Again, ~. :B and C are the intercepts with the x, y and z axes.

The Milier indices of the plane are now: h = 1/ ~. k = 1/ :Band I= 1/C, so:

hx+ky+IZ = 1

The only rule is that h, k and I must be integer numbers again. The notation is then: these inlegers between round brackets, with negative numbers indicated by an overbar.

Note that we can nat calculate ~. '13 and C from h, k and /: the rules given here are one-way, because the intercepts belang to one specific plane, while the Milier indices belang to an infinite family of planes (one in each unit cell). The reverse processcan only be done for one specific plane (e.g. from a drawing): delermine the intercepts with the axes, and calculate h, k and I trom them.

52 Materia/s Science for Biomedica/ Engineering

In Figure 4.17, several planes are shown, tagether with the appropriate choice of origin (black dot) for determining the Milier indices.

z • I

(111)

/

- --

/ , ~t~-'

x

z

• /l I

I I I I I I I I

--- __ .,. y

----, ·----· y , , , , ,

-(110)

x

z

•

z .. I

(011)

(142)

Figure 4.17. Several planes with their Milier indices. Note that the origin is not always in the same location.

Also here, a collection of symmetrically identical planes can be bundled together, in this case with the notation {hkl}.

Example 1: All side planes of the cubic unit cell can be colleeled as: {1 00} = (1 00), (01 0), (001 ), (1 00), (0 1 0), (00 1)

... .... . ..


Example 2: All diagonal planes together are indicated as {11 0}:

.. .:..··············::-.... ::...L--L.--

············

~~+"; ----7""./"--11 ~I / I

I ~ I I : I

I : I

: I I .... •············i·t········ ······

\ \

.......... )'············ .......... .

53

The advantage of this notatien with Milier indices for planes is that there are two very convenient mathematica! relationships now: - The vector that is normal to a plane (hkl) is simply [hk~: this vector has an angle of 90° with all the veetors in the plane (hkl). Check this with the dot product! - The spacing (distance) between all the planes with indices (hkl) in the various unit cells of a cubic crystallattice with a cube edge a is given by:

We can now also find out which planes are the most densely packed with atoms in a crystal. We will see later that these planes are important for deformations of the material: BCC-(11 0); FCC-(111 ); HCP-(0001 ).

Intenne.z.zo For hexagonal unit cells, 4 numbers are used to indicate directions and planes. These Miller-Bravais indices are due to a different choice of unit vectors. Directions: [uvtw] instead of [u'v'w1, with u= n(2u'- v')/3, v = n(2v'- u')/3, t =-(u+ v), w = nw', and choose n so that the smallest integers are obtained. Planes: (hkil) insteadof (hkl), with i= -(h + k).

I ntenne.z.zo For the study of crystal structures, we can use various techniques. The most often used is X-ray diffraction: this gives the spacing d between planes of atoms, when X-rays with wavelength À are diffracted from a crystal under an angle 8:

n'A = 2dsin e lndividual atoms can beseen with field ion microscopy, scanning tunneling microscopy (STM), or atomie force microscopy (AFM).


4.6 Glasses

When the crystalline order is nat present. e.g. by very rapid cooling of a malt, we gat a disordered solid: a glass. lnorganic glassas are not very viscous liquids, as is somatimes said: they are just as solid as crystalline materials with strong bands, but they lack order (see Figure 4.18).

Figure 4.18. G/asses havenoorder intheir covalent bands (left), in contrasttoa crystal/ine substance (right).

Glassas can be found for metals (metallic glasses), but mostly we see this for caramie oxides. They can be completely glassy, or a mixture: the glass-ceramics.

Some polymers at low temperatures are actually also glasses, but their behaviour is so different that we will treat them separately.

Most oxide glassas are based on silicon oxide (Si02, sand), but various other oxides can be mixed in to change the properties of the glass. Examples are: AI203 (fibreglass), 820 3 (electrical applications), Na20 (lamp bulbs), CaO (fibreglass), and PbO (lamp tubing, crystal glass ).


4.6.1 Gtass temperafure

The difference between the glassy and crystalline states of a material can be seen most easily when the volume is foliowed as a tunetion of temperature during cooling (see Figure 4.19).

3

2

1 Crystal

I ---M

Transformation range

I .....____ I

Supercocled liquid

Temperature

Figure 4.19. Whereas crystalline substances (1) show a clear melting point, g/asses (2,3) show a transition at a variabie temperafure (2: slow cooling, 3: rapid cooling).

A crystalline material solidifies at one specific temperature (melting temperature, Tm). with a clear decreasein volume. For glasses, the liquid melt is first supercooled, and then shows a transition to the glassy state at the glass temperature (T9). This glass temperature is dependent on the cooling rate: taster cooling (3 in Figure 4.19) gives éihigherT9 . The volume does deèrease during supercooling and glass formation, but glasses always have a larger volume than crystalline solids, because the packing of atoms is not so efficient.


Problems

4.1.a. Caesium metal has a BCC structure with a lattice parameter of 0.6080 nm. Calculate the atomie radius.

b. Thorium metal has an FCC structure with a lattice parameter of 0.5085 nm. Calculate the atomie radius.

4.2.a. Rhodium has a lattice parameter of 0.3805 nm and the atomie radius is 0.134 nm. Calculate from this whether this metal has a BCC or an FCC structure.

b. Niobium has a lattice parameter of 0.3307 nm and the atomie radius is 0.147 nm. Calculate from this whether this metal has a BCC or an FCC structure.

4.3. lf the atomie radius of Al is 0.143 nm, calculate the volume of its unit cell in cubic meters.

4.4. Calculate the number of atoms per cubic meter in Al.

4.5. Titanium undergoes a change from HCP to BCC upon healing above 882 oe. Assume that in HCP Ti, a = 0.295 nm and c = 0.468 nm, while in BCC Ti, a = 0.332 nm and 1 atom/point. What is the fractional volume change when HCP Ti transfarms to BCC Ti?

4.6. Calculate the percentage volume change when FCC (y) iron (a= 0.365 nm) transfarms to BCC (ö) iron (a= 0.293 nm) at 1394 °C.

4.7. Calculate the theoretica! density of beryllium from the tact that it is HCP with a :;;;: 0.2286 nm and c = 0.3584 nm.

4.8. CaO has a rocksalt structure with a lattice parameter of 0.480 nm. Delermine the theoretica! density of CaO and the number of atoms per unit cell.

4.9. CdS has a cubic unit cell, and from X-ray diffraction data it is known that the cell edge length is 0.582 nm. a. lfthe measured density is 4.82 g/cm3

, how many Cd2+ and s2

- ions are there per unit cell?

b. Based on your answer in a., what type of cubic structure is present in CdS?

4.1 0. What are the Milier indices of the directions 1, 2, and 3 in the Figure below?

4 lnorganic Materlafs I

4.11. What are the Milier indices of the planes a, b, and c in the Figures below?

' ' ' ' ' ' '

c

' ' ' \ \

\

4.12. Select all of the directions that lie in the (111) plane of a cubic crystal: a. [111] f. [1 01]

b.[111] g. [321]

c. [100] h. [211]

d. [110] i. [523]

e.[112] j. [102]

57

4 .13.a. Delermine the Milier indices of the plane that passes through the three points (0,0, 1 ), (Y:!, 1 ,Y:!), and (1 ,%, Y:!) within a cubic lattice.

b. What are the coordinates of the intereepi points on the x, y and z axes? c. What are the Milier indices of the direction connecting the last two points in

problem a.?

4.14.* Calculate the density of a silicon single crystal with a diamond-cubic structure. Does it fit with the experimental value?

4.15. * We are studying a single crystal of chromium (Cr). a. Calculate the radius of Cr from the lattice parameter. Does this fit with the kind of

bond? b. Calculate the number of atoms in the unit cell from the density, and check with the

expected value.

4.16.* Give a vector in the plane (142), and explain how you obtain it.


s lnorganic

Materials 11:

Mechanica I properties and structu ra I defects

Now that we know the structures that are expected for inorganic materials, we can see what the relation is with their mechanica! properties.

We have already looked at the Young's modulus (stiffness} in Chapter 3, so here we will study yield and fracture.

Learning without thought is lost; thought without learning is dangerous.

:fL1ç-=f or Köng Fu-Zi or Köng zi (Master Köng), "Confucius" (551-479 BCE)


5.1 Yield and defects

5.1.1 Yleld and sheartng

Yield occurs when the material is deformed permanently, so we must move atoms by breaking bands and reforming them in another location. In a crystal, this can happen most easily when we move different planes of atoms parallel to each other (shearing). This usually happens along the planes with the highest density of atoms, since then the bands can reform between the planes after only a small movement during the shearing process.

lt is now important to realize that during a tensile test, where we stretch the material with a force perpendicular to the cross section of the test bar, shearing farces still occur. These are the farces that result in yield.

We can see this first in a very simplified, two-dimensional case. This is nota realistic model: do not use this for problems or examinations!

The external force F leadstoa tensile stress: cr = FIA. lnside the material, shearing can occur when we look at a cross section with an angle e to the normal cross section, which is perpendicular to F. The area of this tilted cross section is:

A'= A cose

The force component parallel to this cross section is given by:

F'= Fsina

Therefore we can write fortheshearing force parallel toA':

t = ~: = crcosasina = ,Y,;crsin2e

This shearing force is then maximal for e = 45°:

'<max = cr/2

This value is aften referred to as the tensile yield stress:

cry = 2'<max

Also, one can speak about the shear yield strength:

t = cry/2

F

/'é\,\ .. F'

t

A' ~-:;:;-/ --------------:...----- '

,,::-:::~~-~--------_ .. __________ j

F

However, real materials are three-dimensional, which means that the analysis above is incomplete. What exactly happens at the microlevel is different for materials that consist of a single crystal, and materials that are polycrystalline. We willlook at the single crystals first.

5 lnorganic Materlafs 11

5.1.2 Single crystols

In a single crystal during stress, yield can occur by shearing along planes in the crystal structure that are ciosest to the 45° angle. This indeed happens for singlecrystal materials, as can be seen in Figure 5.1.

Figure 5.1. Stretching a single-crystal materialleads to sliding of layers of the material in roughly 45 o angles.

61

Fora specific crystal structure, we can define a slip system: a combination of a p/ane along which shear will.,<l!ccur, and a direction in that plane in which shear will occur (see Figure 5.2). Thè slip planes will be the most densely packed plane in the crystallattice, and the slip direction will be the most densely packed direction. In that way, the smallest steps have to be made befare movement occurs.

slip direction

F

F

A

slip direction

F

<ls

~'',,

'',,,, .. slip plane

normal

Figure 5.2. The slip system in a single crystalline material.


We now have two different angles: as between the applied force and the slip plane (or rather, the normal vector of the plane), and J3s between the force and the slip direction. as is the same as 9 In the two-dimensionat case, so we have the same formula for A', but J3s is new (it is the component of the force along the slip direction ). The angles are in different planes, so they can independently assume any value. From the Milier indices of the slip plane normal, the slip direction, and the applied force we can calculate the cosines of as and J3s with the dot product (see section 4.5).

The internal shear stress intheslip system (•) canthen be written as a tunetion of the externally applied stress (a= FIA):

't FcosJ3 __ .....:.....:s:._ = O'COSU

5 COSJ3

5 A/cosa 5

Fora specific single-crystal material, a critica! shear stress ('te) can be measured. This is the stress where yield first occurs in a slip system:

Oy cosa 5 cosJ3s

Example: Silver has an FCC structure, where the slip system is {111} <11 0>, with a critica! stress •e = 0.58 MPa.

See Table 3 in Appendix E for an overview of slip systems and critica! stresses. In genera!, FCC structures have more slip systems than BCC or HCP, and therefore materials with an FCC crystal structure will be more ductile.

What can we now predict theoretically forthese critica I shearing forces? What must happen during yield is the sliding of layers of atoms over each other (see Figure 5.3). The first step is the (elastic) stretching of the bonds between the atoms to "lift up" the top layer, so it can move. This is the difficult step, so it will delermine the critica! stress.

R -

shear stress

displacement, x

Figure 5.3. A simpte model for plastic deformation in a perfect crystal.

5 /norganic Materlafs 11

For atoms with an interatomie distance Rand layers with spacing d, a small displacement x winlead to a shearing stress:

Gx t = Gy = d (with: y ""tan(y) = xld)

As a simple model we can now write for the shear stress during the movement:

For small values of x we can write:

2nx

R

Then we can calculate for the theoretically expected shear yield strength:

GR t =-

max 2nd

63

We now have a prediction that the yield strength is related to an elastic property: the shear modulus. (This is why insection 2.4 we gave yield strength as av/E)

Example: In an FCC crystal with edge a, and 1 atom per point, we find that the atoms have a distance R = a/...J2. Fora (111 )-plane in this crystal we can calculate the spacing d = a/...J3, so we then predict: tmax"" 0.2G.

When we look e.g. at silver (G = 19.7 GPa), we would now predict: 'max = 4 GPa, so the yield stress would then be av = 8 GPa.

A more sophisticated model for the torces between the atoms prediets a somewhat lower value for the shear yield strengthof silver: tmax = 0.77 GPa.

But the experimental value for the critica! shear stress of silver along the (111 )plane is 3 orders of magnitude lower: 0.00058 GPa. Th is is very typical, as can be seen from these examples (first predieled and then measured values):

Al 900 vs. 1 MPa Cu 1200 vs. 0.7 MPa Zn 1300 vs. 0.2 MPa a-Fe 6900 vs. 28 MPa

This shows that there must be defects in the ideal crystal structure that delermine the mechanica! properties much more than the ideal crystal structure.

64 Matenals Science tor Biomedical Engineering

5.1.3 Defects

There are lots of point defects (zero-dimensional) that can occur in a crystal structure. In Figure 5.4 we see several of them: - vacancies: missing atoms in the lattice - interstitials: additional atoms inserted in the lattice - impurities: atoms that are replaced by others - ionic defects: Schottky and Frenkel defects, where the overall charge must remain neutra I.

interstitial

vacancy

impurity atoms

Figure 5.4. Point defects in a crystallattice: vacancies, interstitials and replacements (left); Schottky and Frenkel defects (top right and bottorn right respective/y).

More important for the machanical properties are the line defects (one-dimensional), or dislocations. There are two types of dislocation: edge and screw, as can be seen in Figure 5.5.

Figure 5.5. One-dimensional defects can be present as linear edge dislocations, or helical screw dislocations.

5 lnorganic Matenals 11 65

In addition, in polycrystalline materials there are the places where the crystals (grains) stick together: the grain boundaries. These are planar (two-dimensional) defects. The structure of the grain boundaries is often called the microstructure or morphology of the material, which is strongly dependent on the processing.

Finally, when all crystalline order is gone, we have a three-dimensional defect: a glass.

So, a material can have different types of defects: - zero-dimensional = point defects - one-dimensional = line defects (dislocations) - two-dimensional = plane defects (grain boundaries) - three-dimensional =volume defects (glasses).

Intermezzo How can one see defects?

• The presence of dislocations, and the microstructure or morphology can · with various techniques: optical microscopy, SEM, TEM, and AFM. TEM can show lines of dislocations, electron microscopy and optical microscopy can show grain boundaries.

Of course, one always has to look at various length scales when considering the effects of defects on the machanical properties of a materiaL

! Typical length scale Structural feature

~:.m~~---------4-G-r-a-in--st-ru_c_t_u-re-----4~~----~ 10 ~-tm

10 nm

sA

Single grain

Dislocation cells

lndividual dislocations

Atoms I crystal structure


5.1.4 Mobility of disfocations

Oislocations are especially important for the mechanica! properties of crystalline materials, since they can easily move through the crystallattice, thereby causing yield. For the movement of a dislocation, fewer bands have to be braken and remade at the same time, compared to movement of planes of atoms (see Figure 5.6).

1

a

1

T

a b c d e a b c d e

1 2 3 4 1 2 3 4

b c d e

2 3 4

Figure 5.6. Movement of dislocations due to shear stress leads to plastic deformation.

The ditterenee between the yield of crystalline ceramics (brittle) and crystalline metals (ductile) can be explained mostly by the ditterenee in mobility of dislocations. When a dislocation moves, bands must be braken and reformed with different atoms. For metals, this very easy, because the metallic bond is the sametor all directions (electron sea). In ceramics, the ionic or covalent bonds are dependent on the direction, soit is more difficult to switch from one atom to the next.

In metals, the dislocations are therefore very mobile, soplastic deformation (yield) is very easy: metals yield befare they break, bath in tension and in com:Rression. The typical stress necessary to move dislocations in metals is 10--4E to,1o- E. Since ductile fracture in metals also involves plastic deformation, the tensile stress of metals (crT) is also determined by the mobility of the dislocations: when crv goes up or down, so does crT.

In ceramics the dislocations are much less mobile, and yield is more difficult. Ceramics normally break befare they yield, although in compression yield is possible, since the material cannot break so easily then. (Ouring bending, there are bath lension and compression, and ceramics will then fail on the stretched side.) The typical stress necessary for moving dislocations is 10-3E for ionic bonds, and 10-3E to 1 o-2E tor covalent bands. The covalent network ceramics are therefore the most brittie (highest cry). Fracture is usually brittie (see section 5.2).

So, we see the following very important relationship between micro-, meso- and macroscale:

Directionality of bonds ~ Mobility of dislocations~ Yield stress T ensile stress ( metals)


5.1.5 Microscopie anisotropy vs. macroscopie isotropy

We now come back again to the difference between single crystals and polycrystalline materials. Single crystals are usually anisotropic: they have different machanical properties in different directions. In polycrystalline materials, we see an averaged property: the material is isotropic.

Example: BCC a-Fe has different Young's moduli in different directions: E[o01J = 125 GPa, E11 11J = 273 GPa, and E[110J = 210 GPa. Polycrystalline a-Fe has a modulus E = 210 GPa in all directions.

Also for the yield stress, we see that the grain boundaries are the weak spots of a polycrystalline substance: instead of the critica! shear stress in a slip system, we havetolook at the shear stress where the grain boundaries start sliding: 'tgb·

Although there are a lot of grain boundaries in a polycrystalline material with all kinds of orîentations, the shear will be strongest along the 45° angle, so these grain boundaries will move first. However, the movement of dislocations usually dominatas the yield process.


5.2 BriHie fracture and crack growth

When a material breaks, atomie bands are braken, and new surfaces are created. lt should therefore be possible to understand tensile strength in terms of interactions between atoms. Since the behaviour of surfaces is also very important for biocompatibility, we shall treat it in some more detail first.

5.2.1 Surface fension

The atoms on surfaces have a higher energy than the atoms in the bulk, since they lack bands withother atoms that atoms in the bulk have (see Figure 5.7).

Figure 5. 7. Atoms at the surface miss bonds compared to bulk atoms.

Therefore there is always an excess surface energy (on the order of 1-10 J/m2). A

material will therefore always try to minimize its surface area if possible. Of course, the surface energy depends also on the medium (air, water) with which

the surface makes contact: when the atoms on the surface have a good interaction (bands) with the molecules in the medium, the surface energy will be lower. In most tables, the surface energy is given with air as the medium ( almast no interaction possible}, but in general the surface energy should refer to an interface between two materials.

The excess surface energy is called the surface tension y. The unit for y can be as for energy [J/m2

], or as for force [N/m] (these units are identical). The interaction with liquids (water, blood) is very important for biomedical

materials, so we wil I introduce some properties related to this. We will see in section 6.1 that we can use these properties also in more general situations. When a drop of liquid (L) sits on asolid surface (S) in the presence of air (G), the contact angle eis determined by the surface tensions YsG. YLG. and YsL (see Figure 5.8).

gas

YsG YsL solid Figure 5.8. A drop of liquid on a surface has a contact angle that is determined by

the surface tensions.

5 lnorganic Materlafs 11

This experiment can therefore be used to delermine one of the surface tensions, when the two others are known:

YsG = YLs + YLG case

One speaks about wetting of the surface (film formation) when e = 0°, and dewetting ( formation of drops) when e = 180°. For wetting one should have:

YsG = YLs + YLG

For dewetting one should have:

The cohesion of a material A is the work necessary to create two new surfaces (usually relativa to air, but this may be different in a biomedical case):

Wc,A =2yA

69

The adhesion between two matenals A and B is the work necessary to break open a common interface:

To see whether a material B will farm a film on a surface of material A, it is useful to calculate the spreading coefficient of B on A:

SBA = Wa,AB wc,B YA YB- YAB

lf SsA > 0, a film of B will be formed on the surface of A, while for SBA < 0 drops of B will be formed on A.

Example: Water has y = 72.75 mJ/m2, and n-octane has y = 21.8 mJ/m2 (bath relativa

to air). The interface energy between water and n-octane is 50.8 mJ/m2. For n-octane

on water we now find: Sow = 0.15 mJ/m2 > 0, so n-octane will form a film on water.

When a mixture of materialsis used (e.g. alloys of metals), we must realize that the component with the lowest y will be more concentraled on the surface, since then the total surface energy of the material will be lower. Therefore, the bulk composition is nat always the same as the surface composition. This is very important for biomedical applications. Of course. the surface atoms can also react more easily with the environment: titanium is e.g. always covered with a layer of Ti02 due to reaction with air. Th is layer will behave quite differently than bulk Ti.


5.2.2 Orowan model

There is a simple model (the Orowan model) to estimate the tensile stress of a material from its surface energy. As we saw before, brittie fracture involves the creation of two new surfaces, sa it casts energy:

Etract = 2yA

We would like to express this in terms of stress and strain. We assume a perfectly crystalline material, and want to separate two layers of atoms, which are perpendicular to the stress. Normally the atoms are situated at a distance Ro, and this has to be stretched by an additional distance R1 befare the bond can be braken (see Figure 5.9). The necessary stress is approximated with a sine function.

R, Cf ...

O'max

Rot Cf

1 Distance, R

Cf

Figure 5.9. Separating two layers of atoms wil/lead to an increasing stress.

As can be seen in Appendix M, the Orowan model yields a maximum stress (which is also the tensile stress) given by:

Experimentally we aften find that R1 ""R0, sa y"" ERoht2 (see Appendix M), and we then have a tensile stress:

E E crT ;:::;-;:::;-

1t 3

This is the reason that tensile strength is aften given as the ratio <JTIE: from this theory we expect this to be the same value for all crystalline materials.

We can use better models for the fmces between atoms, which lead to somewhat lower values for the tensile stress: E/8 to E/15.

However, all these predictions are far too high: for iron we have E = 211 GPa, which prediets that crT = 14-67 GPa, while we find experimentally that <JT = 0.1 GPa for bulk iron. For iron whiskers (smal! needle-shaped single crystals) we find a much higher tensile strength (13 GPa), but it is stilllower than predicted. Obviously, sarnething is wrong. In genera!, a higher tensile strength is found for fibres and whiskers, as shown in Figure 5.10.

30

20

10

+- .. bulk

5 lnorganic Matenals 11

... .. continuous - Caramies

fibers - Metals

IIIIIIGiass

Figure 5.10. The tensile strength of materlafs depends on their dimensions.

71

The samewas found by Griffithand others (e.g. Gordon) for fracture in glass fibres: the tensile strength is larger when the diameter is smaller (especially under 10 flm). lt was observed that the fracture was caused by small cracks on the surface of the glass, which needed a minimum size to start growing. Because during the growth of the cracks only a few bands were braken at the same time, a low stress was enough for complete fracture. Below a certain diameter of the glass fibre, the cracks were toa small to start growing, and fracture only happened when the bonds between the atoms broke all at the same time, just as in the Orowan model. By extrapolation to very low diameter (see Figure 5.11 ), Griffith indeed found a tensile strength of 11 GPa, which nicely agrees with the theoretica! value of 14 GPa for the material he used. For quite thick fibres, he found the same strength as for bulk glass: 170 MPa.

co a. Q .&:. -Cl c: ~ 1ii ..!!! ïii c: ~

4

3

2

5 10 15 20 Thickness of fibre/J.im

Figure 5.11. The tensile strengthof g/ass fibres is inversely proportionaf to the diameter.


For whiskers of Si and ZnO a similar relation between tensile strength and diameter was found. Since whiskers are perfect single crystals, we do not have surface cracks here, but fraeture starts at the "steps" on the surface where new layers of atoms start. When the surface of a material was made so smooth that steps do not occur, the tensile strength went up in value. For normal, polycrystalline materials the cracks can start on the grain boundaries, which are weak spots.

So, the tensile strength of a material is completely determined by the possibility of small cracks or imperfections to start growing.

5.2.3 Stress concentrations

When analyzing the growth of crack in a material, it was found by lnglis that there must be stress concentrations at the end of the cracks, partly because the stress through the material is carried by a smaller surface. This increased stress leads to an unexpectedly quick plastic deformation (yield) in metals, or easier fracture in ceramics.

In Figure 5.12, cis the depth of a crack on the surface, or it is half of the length of an interior crack. Pc is the radius of a circle that fits in the tip of the crack (the radius of curvature): a sharp crack has a small Pc, and a blunt crack has a large Pc· The stress applied that is from the outside is cra = F!Ao, but the real stress at the tip of the crack (crint) is much higher:

When the crack is exactly circular (as seen from the side), then Pc= c, and O"int = 2cra. For most cracks, however, the shape is elliptical with a sharp tip, and Pc << c, so O"int >> cra.

Pc ' ' ' I

.. --- ...

-----------------------\~ ... I ... _!,

These stress concentrations can lead to crack blunting in metals: plastic deformation changes the shape of the crack tip, thus lowering the interna I stress. However, in caramies the stress concentrations are expected to cause growth of the cracks, and materials would break very easily. This is obviously not always true, and Griffith proposed a model to show why this is not the case.

5 lnorganic Materlafs 11 73

i i F i

<J\

V ~ '-..... r-

~ __..

2c

(J \

~ 1"-1-·

c

! ! F ! Figure 5. 12. Extemal or internat cracks in a materiallead to much higher internat

stresses at the crack tips.

5.2.4 Griffith model tor brittie fracture

Griffith realized that the energy associated with a crack is determined by two terms: (i) the energy increases because two new surfaces are created, and the longer the crack, the more surface there is; . (ii) in an area around the crack-except the tips è.ircourse-the material is no longer stressed, and elastic energy is released.

In a plate of unit thickness (see Figure 5.13), the new surface are created by an interna I crack of length 2c is 4c (both sides of the crack), and the area that relaxes around the crack was found to be ca. 2na2. This then leads to the following expression for the total energy of the crack:

4cy [J/m]


Crack length

-1tC2 t::l-1 E

Figure 5.13. A growing crack first increases in energy, but then decreasas again due to stress re/axation in the shaded area around the crack.

This means that the growing crack first increases in energy, and only starts growing spontaneously (decreasing energy), when a certain critica! crack length is passed.

The critical crack length is reached for the maximum energy (dEcrack/dc = 0). This gives for the critica! length of the cracks at a certain applied stress cr:

When cracks are larger than c*, the stress cr willlead to fracture. Another way to look at this formula is to calculate the minimum applied stress

needed to make a crack of length c start to grow: the critica! stress.

cr* = ~2Ey [Pa] 1tC

Now, when the applied stress is larger than cr*, the cracks wilt start growing, and the material wilt break.

In order to be able to talk about crack length and stress at the same time, the stress intensity Kis defined as:

To combine the material properties E and y in one number, the critica! stress intensity factor or fracture toughness K1c is defined as:

As the name already suggests, K1c is a measure for the toughness of a material, which is aften used besides the area under the strass-strain curve.

5 lnorganic Materlafs IJ

The criterion for fracture is now very simple:

Fracture can be due to a too high stress, or too long cracks, and the critica! values can now be written as:

cr* =

5.2.5 Ceromics vs. metals

since we get some compli. However, this farm of the

75

As seen in Figure 5.14, growth of cracks in ceramics occurs by breaking bands at the tip of the crack. All bands have the ideal strength (as predieled by the Orowan model), and this is reached by stress concentration.

c

Atoms peel apart at E/15

Figure 5.14. Growth of a crack in a ceramic must break atomie bonds.

Therefore, the value of K1c is indeed as expected from the value of E and a surface energy of a few J/m2

• Griffith's model works very well here:

Example: For Al20 3 we have E = 390 GPa and K1c = 3.7 MPa·m112• This gives y = • 1 q 1b

J/m2. The tensile strengthof Al20 3 is ca. 270 MPa, so there must be cracks with a

maximum length of ca. 60 ,....m.

\


For metals, K1c is much higher than expected from the values of E and y. Something is missing here from Griffith's model. The difference between metals and caramies is that metars can show yield: plastic deformation, at much lower stresses that the stress needed for breaking metallic bonds. This plastic deformation absorbs a lot of energy because bonds have to be broken before they are formed again (see Figure 5.15). This means that the area around the crack can absorb more energy than expected from just crack growth. Therefore, it costs more energy to open up a surface than just the surface energy.

In metals the cracks grow by ductile tearing: the material is torn open, and a large area around the crack must be plastically deformed. Therefore, the yield strength is more important than the theoretica! tensile strength. This ll~ha~iour-ran be incorporated in Griffith's model by substituting the}Wörk-of fracture,{IM) for the surface energy y: " I

For ceramics we find that tM = y, but for metals we have Wt >> y.

c

Plastic zone

. ..

Figure 5.15. In metals, there is plastic disfortion at the tip of a growing crack before bonds break.

Intermezzo Be careful: 2Wt is somelimes referred to as Ge, the critica! strain energy release rate, or just toughness. This last term is very confusing, si nee toughness is already used for K1c itself. In Chapter 2, the normalized toughness that is plotled is Ge lP in Jm/kg. Other books simply give y another value, without saying that it is no longer just the surface energy. And other books talk about Yc + YP• where the first term is the surface energy, while the second term includes all energy necessary for plastic deformation.

5 lnorganic Materia/s 11 77

Example: For aluminium we find E = 72 GPa, and K1c = 45 MPa·m 112. This leads to

W1 = 14 kJ/m2. This is indeed much larger than the few J/m2 that are expected for y.

As has been mentioned before, the ductilit~ of metals is due to the high mobility of dislocations. When a metal is sheared, ca. 10 6 dislocations are formed per m2

. Since each dislocation has an energy of ca. 1 o-9 J/m, this means that ca. 107 J/m3 is necessary for creating the dislocations. When an area with a thickness of 1 cm around a fracture has to be deformed, this results in a workof fracture of ca. 105

J/m2. This is in good agreement with the numbers found from the fracture toughness.

Of course, the toughness is dependent on the temperature, as is the yield strength. In Figure 5.16, we can see the dependenee for stainless steel.

~ E ro a.. ~ -... (/) (/) Q) c

..r:::. Cl :::::1 .8 ~ :::::1 t5 e:! u.

200

160

120

80

40

0

-150

1200

I 1000

V Yield strength I - 800

ro - I

I a.. ~ -...

600 ..r:::. -Cl c ~ -/ Fracture

,./"" toughness

400 (/)

"C a; >=

--- ----

200

0

-100 -50 0 50 100

Testing temperature/°C

Figure 5.16. The mechanica/ properties of a metal depend strongly on the temperature.

Intermezzo Fatigue When a material is stressed and released repeatedly, or stressed periodically, there will also be fracture aftera certain time, because plastic deformation occurs that leads to crack growth until c* is reached. We will nat look at this in detail here.


5.2.6 Overview

For inorganic materials, there are several relations between structure and mechanica! properties on micro-, meso- and macro-level. They are given schematically in Figure 5.17. Ceramics will always show brittie fracture, and metals can undergo both ductile and brittie fracture, depending on whether crr or a* is the lowest value.

Micro Meso Macro

Bond ~----------- E, stiffness

strength ----------- y, surface

energy ~

Direction dependenee afbond

. . metals: caramies Mob1hty of via Wr

dislocations~ ::~~::ghness :; crr, ductile fracture

Lengthof microcracks cr*, brittie fracture

Figure 5.17. Relations between micro, meso and macro in inorganic materia/s.

5.3 Glasses

Glasses are inorganic materials without crystalline order. Yield in glasses is caused mostly by viseaus flow: the atomie bonds are broken

and reformed in a random manner, causing plastic deformation. The bonds in glasses are as strong as in crystalline ceramics, so yield is difficult and most glasses break befare they yield (at least, at low temperatures).

Fracture in glasses follows the same mechanisms as in crystalline solids: it is determined by the Griffith model with the growth of small cracks and a fracture toughness, which is low for ceramic glasses. So, glasses are not very tough.


Problems

5.1. A single crystal of Mo, shaped as a cylindrical rod with a 1 cm2 cross section, is pulled in tension. lf the crystal axis lies along the [123] direction, at what applied load will crystal slip occur on the (110) plane in the [111] direction?

5.2. A copper sphere weighs 10 g. The surface energy of copper is 1.4 J/m2.

a. Calculate the energy of the total surface of the sphere. b. When the original sphere is braken down into fine spherical grains, each 100 nm

in diameter, what is then the energy of the total surface?

5.3. When liquid mercury (Hg) is put on glass, drops are formed with a contact angle of 140°. lt is known that the surface energy of Hg is 0.47 J/m2

, and that of glass is 0.30 J/m2

•

a. Calculate the surface tension between glass and Hg. b. Calculate the cohesion of both glass and Hg, and explain the ditterenee via the

type of chemica! bands. c. Calculate the adhesion between Hg and glass, and the spreading coefficient of

Hg on glass.

5.4.a. A steel with a yield stress of 1000 MPa contains a circular surface crack. What stress can be safely applied without deformation occurring near the crack?

b. In the same steel an elliptical surface crack has a radius of curvature equal to 0.5 mm. How deep a crack can be tolerated without plastic deformation if the applied stress is 240 MPa?

5.5. A plate of steel has a yield stress of 1 000 MPa. The plate fractured when the tensile stress reached 800 MPa, and it was hypothesized that a surface crack is present. lf the fracture toughness for this steel is 60 MPa·m 112

, approximately what size of crack must be present?

5.6. Soda-lime glass has a surface energy of 0.30 J/m2 and a Young's modulus of 69 GPa. Compute the critica! stress required for the propagation of a surface crack of length 0.05 mm.

5.7. A specimen of a 4340 steel alloy with K1c = 45 MPa·m 112 is exposed toa stress of 1 GPa. Will this specimen experience fracture if it is known that the largast surface crack is 0. 75 mm long? Why or why not?

5.8.a. Compute the crack tip radius for an Ab03 specimen that experiences tensile fracture at an applied stress of 275 MPa. Assume a critica! surface crack length of 0.002 mm and a theoretica I fracture strength of E/1 0.

b. What critica! fracture stress do you expect for this fracture length based on the Griffith model?


5.9.* A certain type of stabilized zirconia (Zr02) is evaluated as biomaterial. The supplier tells us there are internal defects with a lengthof 0.26 mm, and that K1c = 12 MPa·m112

• The Young's modurus of Zr~ is 200 GPa. a. Calculate the maximum stress this material can stand in a tensile test. b. What surface tension can you calculate for this material? Is this an expected

value? lf not, what might explain the deviation? c. Explain why the supplier did not give us information about yield in this materiaL

5.1 0. * Stainless steel (type 304, annealed) is subjected to a tensile test. a. Calculate the maximum engineering stress this material can withstand. Carefully

explain all assumptions. b. What is the maximum elastic strain for this material? Explain why it is so much

smaller than the maximum engineering strain. c. Explain why this material has such a high ductility.

5.11. * A silicon (Si) single crystal is subjected to tensile stress in the [231] direction. What percentage of this stress will be a shear stress in the crystal?

5.12. * A single crystal of chromium (Cr) is stretched in the [120] or [0 11] direction until plastic deformation occurs. Along which direction can the material withstand a higher force?

6 Organic

Materials I:

Structure of (bio )polymers

Organic materials are built up from very large molecules, with only weak bonds between them. Therefore, the materia Is are mostly amorphous, and the molecules are randomly coiled. We can then describe the structure of the material with a statistica I distribution.

When we deform these materials, changing the shape and orientation of the macromolecules is very important.

I cannot teach anybody anything, I can only make them think.

L:roKpUtl']ç (S<Krates) ( 469-399 BCE)


6.1 Overview

6.1.1 Linear polymers ond networks

There are various types of polymers as far astheir chemistry is concerned. We will only briefly summarize this hare; an overview of polymer structures can be found in Appendix I.

Macromolecules are formed by polymerization reactions, of which radical polymerization and polycondensation are the most important ones.

Radical polymerization leads to C-C bonds, and with the usual monomar CH2=CHR, we get as structure for the macromolecule: -(CH2-CHR}n-. Also, other structures are used, such as -(CHrCR'R")0-.

Examples: R = H: poly(ethene}, or poly(ethylene}, PE R = CH3: poly(propene}, or poly(propylene), PP R = C6Hs: poly(styrene}, PS R =Cl: poly(vinyl chloride}, PVC R' = CH3, R" = COOCH3: poly(methyl methacrylate), Plexiglass, PMMA -(CF2-CF2}-, poly(tetrafluorethene), Teflon, PTFE

PP= H -CH-C-

2 I CH3

PS=

Polycondensation leads to the formation of esters, amides and other bonds. The monomars must contain functional groups, such as carboxylic acids, alcohols, amines, etc.

Exampfes: -(NH-R-CO)n-. nylons or poly(amide)s -(0-R-CO}n-. poly(ester}s -(0-R-NH-CO)n-. poly(urea)s -(0-R-0-CO)n. poly(carbonate)s

When different monomars are mixed, we get copolymers. Basedon the sequence of the monomers, we get different types, as shown in Figure 6.1 .

•••••••••••••••••••

•••••••••••••••••••

••••••••••••••••••• Flgure 6.1. Copolymers can be altemating, random, and block (/eft, top to bottom);

other types are graft copo/ymers and terpo/ymers (right).

6 Organic Materlafs I

The properties of polymerie materials can be tuned by changing the mixture of monomers.

83

For many polymers, there are different forms based on the stereochemistry of the carbon atoms in the different units: the contiguration can be the same in all units (isotactic, see Figure 6.2 left), it can alternate in subsequent units (syndiotactic, middle), or be completely random (atactic, right). This has important consequences tor the way in which these polymers can form crystals.

Figure 6.2. Different stereochemistries in polymer chains: isofactie (/eft), syndiotactic (middle), and atactic (right).

6.1 .2 Modern architectures tor polymers

Besides the standard linear polymers and networks there are nowadays also polymers with other topologies, each with specific properties. Branched polymers do not consist of a single linear molecule, but contain rnany places where the polymer chain branches. We have already seen the graft copolyrners (see Figure 6.1 ). Other exarnples are: brush-, star-, and hyperbranched (arborescent) polyrners, dendrirners, and LC (liquid crystalline) polymers (see Figure 6.3).

Figure 6.3. Po/ymer architectures: brush, star, hyperbranched, and dendritic.


6.1.3 Crosslink density

We are usually interestad in the way in which the macromolecules are packed together to farm a materiaL This can be influenced by covalent bands between polymer chains (crosslinks), which lead to the formation of a network. This gives another way of classifying the polymers: the amount of crosslinks that are present.

Reversible therrnoplasts are amorphous coils of macromolecules with possibly some crystalline areas (see Figure 6.4). There are (almost) no covalent crosslinks, so the material can be deformed irreversibly. However, at low temperatures the crystalline regions in the amorphous matrix can tunetion as crosslinkers. And even for completely amorphous polymers at very high molar masses (very long molecules), there are entanglements between the molecules that can act as crosslinkers. But the crosslink density is low in all cases.

Chain

branching

Amorphous region

Figure 6.4. Typical structure of a thermoplastic polymer.

Examples: Non-crosslinked macromolecules, such as PE, PP, PVC, nylons.

Thermoharders are compact networks of covalent bands between macromolecules, which are formed irreversibly, e.g. by healing or UV-irradiation (see Figure 6.5). Once formed, the network cannot change shape anymore. There is a very high crosslink density. These network compounds are aften called resins.

Figure 6.5. Typical structure fora thermoharder.

6 Organic Materials I

Examples: - Network structures based e.g. on reaction between formaldehyde and phenol (Bakelite, see below), ar formaldehyde and melamine:

>~o.ó- OH OH OH OH

85

- Epoxy resin, -(O-Ph-C(CH3)2)-Ph-O-CHrCH(OH)-CH2)n-, with compounds that conneet the OH-groups (crosslinkers). - Polyester resin, -(CO-(CH2)m-CO-O-C(CH20H)2)n-, also with crosslinkers.

Elastic rubbers ar elastomers are networks with a small number of covalent bands (see Figure 6.6), which are still elastic. The crosslink density is nat very high.

Figure 6.6. Typica/ structure tor an elastomeric polymer.

Examp/es: - SBR, styrene-butadiene rubber, copolymer -(CH2CHPh)n-(CHrCH=CH-CH2)m-, with crosslinks from e.g. sulphur between the C=C bands. - NBR, nitrile-butadiene rubber, copolymer -(CH2CHCN)n-(CHrCH=CH-CH2)m-, also with crosslinks from e.g. sulphur between the C=C bands. - Poly(isoprene), natural rubber, -(CHrC(CH3)=CH-CH2)n-, crosslinked with sulphur - Silicone rubber, -(Si(CH3)2-0)n-, crosslinked via reactions with peroxides.

Intermezzo Elastomers have high elasticity, but they can nat be melted and shaped after synthesis. Thermoplastscan be melted, which has advantages for processing. Thermoplastic elastomers combine the properties because there are reversible crosslinks e.g. by ionic interactions ar hydragen bonding. At high temperatures these crosslinkscan be opened, and the material can be processed. At low temperatures there is high strength and elasticity due to the crosslinks. An example is Lycra (Spandex), which contains many urea groups (-1\JH-CO-NH-) that can farm strong hydragen bands, while there arealso long ethylene oxide chains (-CHrCH2-0-)n that are very flexible.


6.1.4 Molar mass distribution

Most synthetic polymers are a mixture of macromolecules with different lengths, or degrees of polymerization (DP: number of monomers in the chaln), and therefore different molar masses (also: molecular weight). In order to characterize the whole material, we need an average molar mass. This averaging can be done in two ways that both are important. lt is determined by the way we describe the composition of the mixture of different macromolecu les: with mole fractions or with mass fractions (sometimes, also volume fractions are used).

The number-averaged molar mass (M0 ) is what we would normally consider the average: the molar masses of all macromolecules are added, and the total is divided by the number of molecules. lf we have Nj mol of macromolecules with a length of j monomers (so DP = J), each with molar mass Mo. we get a molar mass Mj = jM0 fora macromolecule with length j.

The total number of macromolecules is:

N 'i.Ni [mol]

We can now define the mole fraction of each macromolecule with a specific length j:

Then, the number-averaged molar mass is given by:

We can also look at the average dagree of polymerization: for one chain it holds that DP = j, and for a mixture of chains we can define a number-averaged dagree of polymerization (the average length):

Th is is the most often occurring molar mass based on molar percentages. Another kind of average can be determined when we look at the mass of a

polymer with a specific length, inslead of the number of chains: this is the weightaveraged molar mass (Mw). There is for polymers of length ja mass present of NjMj kg, and the total mass is the sum of all these masses, so the mass (or weight) fractions of the various types are:

Then we can define the weight-averaged molar mass:

6 Organic Matenals I

This is the most often occurring molar mass based on weight (mass) percentages. We can alsodefine a weight-averaged dagree of polymerization:

The polydispersity of the polymer mixture is now defined as:

L(J 2xJ (L(jxi)l

The more D deviates from 1, the broader the distribution of molar masses is in the mixture, as shown in Figure 6.7.

,_ Q)

E >-0 0. -0 -c: ::I 0

~

Number-average, Mn

! Weight-average, Mw

!

Molecular weight --~

87

Figure 6. 7. The two different averages for polymer molar ma ss. Sametimes Mn and Mw are given with an overbar, to indicate they are averages.

Example: Mix 20 g of a polymer with a molar mass of 20 kg/mol, 50 g with a molar mass of 100 kg/mol, and 30 g with a molar ma ss of 300 kg/mol. The total mass is then 100 g, so the mass fractions are: 0.2, 0.5 en 0.3. Mw canthen be calculated to be 144 kg/mol. The molar quantities follow from Ni = mi /Mj, and are respectively 1 mmol, 0.5 mmol, and 0.1 mmol. Total amount: 1.6 mmol, so the mole fractions are: 10/16, 5/16 en 1/16. This gives: Mn = 62.5 kg/mol. The polydispersity is then: D = MwiMn = 2.3.


6.1.5 Potymers as biomaterials

Polymers are aften used in the human body as sutures, blood vessels, replacements for soft tissues (ear, nose, etc.), and in hip joints. Examples are: - poly(ethylene) (PE, hip, tendon implant, facial implant); - poly(ethylene terephthalate) (PET, aorta, tendon, facial implant)

= -(O-CH2-CHrO-CO-Ph-CO)n-; - poly(methyl methacrylate) (PMMA, Plexiglas, intra-ocular lens, bone cement); - poly(dimethylsiloxane) (PDMS, silicones, braast prostheses) = -(Si(CH3)2-0)n..,.; - poly(urethanes) (PUr, skin and vessels) = -(0-R-0-CO-NH-Ar-NH-CO)n-(various R = alkyl and Ar = aromatic group ). As hydrophilic polymers one aften sees: - poly(ethylene oxide) (PEO = poly(ethylene glycol, PEG)

= -(CH2CH20)n-; ûo - poly(N-vinylpyrrolidinone) (PNVP) = -(CH2CHY)n-. with Y = _ Typical biodegradable polymers are: - poly(lactic acid), PLA, -(O-CH(CH3)-CO)n-; - poly{glycolic acid), PGA, -(0-CHrCO)n-·

Polymers are easy to produce and are flexible, but they can change shape after some time, and they can degrade in a biologica! environment. For many polymers, this last property is actually an advantage, si nee they will then disappear from the body aftera while, when their tunetion is finished. In joints, polymers can sametimes produce particles by friction, which can lead to inflammation.

Some polymers must be modified on the surface in order to become biccompatible or bioinert. Apolar polymers can be treated with e.g. a plasma to get a more polar surface. Also, a more polar hydrophilic polymer (e.g. PEO) can be attached to the surface (graft copolymer).

The surface properties are again related to the surface energy y, which can be determined by measurements of the contact angle with e.g. water (y = 72.8 mJ/m2

versus air). The more polar the surface is, the higher YsG is, and the lower YsL then is (see Table 6.1 ). These values are e.g. of importance when trying to predict whether proteins will attach to the polymer surface or nat.

Tab/e 6.1. Contact angles and surface tensions of various polymers vs. water.

Material

nylon-6 nylon-6,6 PET PS PE PTFE

61" 70" 79" 84° ar

104°

46.6 41.1 36.0 32.4 30.3 20.0

10.8 16.8 22.2 24.8 26.6 37.6


6.2 Morphology: crystalline/amorphous

6.2.1 Polymer crystals

89

In the case of polymers, the molecules are much larger than the unit cells of crystal lattices, so we can only talk about the crystallization of segments of the polymer chain. The chain must then fold up tofarm a regular crystal structure, and we get lamellar shapes of 10-20 nm thickness (see Figure 6.8). These lamellae can farm so-called spherulites.

Figure 6.8. Polymer crystals are platelets with dangling molecules, formed by folded chains.

A polymer can almast never be 100% crystalline, so there is always an amorphous part, and we have semi-crystalline materials. A polymer can normally only crystallize when the structure of the chain is regular (iso- or syndiotactic). Atactic polymers are almast always amorphous.

Biopolymers can also crystallize, althol..lgh this does nat happen a lot in a living organism. Artificial proteins with certain repeating amino-acid sequences can farm lamellar crystals, because ~-sheets are formed between chains with a Gly-Aia repeating unit, while a Gly-Giu pair creates a hairpin turn.

The rnalar mass (and rnalar mass distribution) and the percentage crystalline state determine how the polymer behaves, as shown in Figure 6.9.

100

Brittie

500 2000 5000

Hard plastics

20000 Molecular weight (Nonlinear scale)

Soft plastics

40000

Figure 6.9. Dependenee of behaviour on molar mass and crystallinity.


6.2.2 Glass transition

As we have seen before for inorganic glasses, amorphous polymers also show a glass transition. This glass transition temperature (Tg) is the temperature where segments of the polymer chain start to move, and it is determined by: - chain flexibility, interactions between chains, and crosslinking; less flexibility gives a higher Tg

- chain length, or molar mass: Tg = Tg,oo - ~T n

- the time scale of the measurement; faster maasurement higher Tg.

The cooling rate can then lead to more or less free volume in a V, T-diagram (see Figure 6.1 0): this is the volume ditterenee between the expected volume of the glass above the Tg and the actual behaviour as a melt.

Freevolume

Tg(1) 79(2) Temperature

Figure 6.10. The g/ass temperafure is dependent on the speed of the measurement, and is typica/ for the non-crystalline thermoplasts.

This behaviour is of course typical for the thermoplasts, which have a large amorphous part. Thermoharders are non-crystalline materials, but they do not show a glass transition, because the many crosslinks prevent motion of the main chain. A rubber (elastomer) below its T9 is a glass, with van der Waals bonds besides the covalent crosslinks. Above T9 a rubber is in fact a viseaus liquid, which is kept tagether by a few covalent crosslinks.

Besides a glass transition temperature (Tg) a polymer can also have a melting temperature (Tm) for its crystalline state. Tagether with the molar mass, the temperafure will influence the mechanica! properties of the polymerie material, as shown in Figure 6.11.

6 Organic Materlafs I 91

Mobile liquid Rubber

~ Tm .a T ough plastic ~ (]) Tg a. E (]) 1-

Partially crystalline plastic

101 102 103 104 105 106 107

Molecular weight

Figure 6.11. Both molar mass and temperafure influence the mechanica/ behaviour of polymers.

For various polymers the glass temperature T9 and the melting temperature Tm follow the sametrend (see Figure 6.12).

40 V

30 0

20 0

10 0

0 T,

0

-10 0

PI

PEEK

Tm PPS PEl

PA6 /PE s

POM pp ~PS PE c

AB~PEEK 'PMMA

u

~PS l/VPPS

PET Tg

L!'.fl:~-------------------------------------I

I I pp

~POM BR

PE

Figure 6.12. Trends in Tg and Tm for various polymers.

92 Materials Science tor Biomedical Engineering

6.2.3 Random-coil model tor the amorphous state

In order to describe how an amorphous polymer behaves, we need to know something about its dimensions. Fora polymer that is randomly distributed in space, we can set one end at the origin. The distance to the other chain end can then be called the end-to-end distance re (see Figure 6.13).

z

x

Figure 6.13. The random coil of a polymer chain with the end-to-end distance indicated.

There is a statistica! model (the random-coil model) that is now very useful. This model is also known as the random walk, and it is easiest to first look at it in one dimension: suppose you start on a line at position 0 and you can make random steps to the left or the right of length L. Then your position after one step will be:

In order to get rid of the unknown sign, we will use the square of the distance:

R12 = L2

Now for step N, we can write:

Again we use the trick with the square:

On average, there are as many steps to the left as to the right, so:

(where the <>brackets stand for the average value) and we are left with:

(of course, <L2> = L2, as it is a constant)

6 Organic Matenals I 93

lf you now look back at the starting value of <Rl> (= R12), you can easily see that we

always will get for the so-called root-mean-square value of the distance:

<RN2> = NL2

Amazingly, this is also true when we switch to three dimensions (see Appendix N). lf the polymer chain consists of ns segments, each with length Ls, then the

average radius of the polymer coil is:

Another average radius that is used a lot (based on the distance to the centra of mass) is the radius of gyration, R9. lt is obtained from experimental techniques like light scattering, and for a random coil it can be shown that:

sa:

For real macromolecules, a completely random distribution over space is of course not possible: the bonds have specific angles 9 (according to the VSEPR theory), and there are preferences for certain conformations (staggered) with specific torsional angles 4J. This leads to the expression:

where Cr has a value between 5 and 10, e.g. 6.7 for poly(ethylene), and 10 for poly(styrene ). The 9-factor is 2 for the common angle e = 1 09.5° (tetrahedral). The 4>-factor is often called cr/, where crr varies between 1.5 and 2.5. crr is then referred to as the characteristic radius of the polymer, and is indicative of the stiffness of the polymer chain.

Intermezzo Better theoretica! models predict <r/>- n6

315, and numerical simulations also suggest

<rc2>- ns06, so there are more possibilities to change the formula.

For polymers in solution there can also be an interaction with the solvent, which usually swells the coil, but can also shrink it. Only at a certain temperature for each solvent, the 9-condition, the formula for <rc2> also holds in solution.

~~~~~~~~~~~------


6.2.4 tmp/icatlons for the amorphous state

Suppose we have a poly(ethylene) (PE) with a rnalar mass of 280 kg/mol, and therefore a degree of polymerization (length as number of monomar units) ns = 104

.

For this moleculewethen have: Ls = 0.154 nm and Cr= 6.7, so <rc2> = 1590 nm2.

Then the average distance between the chain ends is 40 nm, and the radius of gyration: R9 ~ 16 nm. ----

The volume ofthis random coil is now( ~(4n/3 <rc2>312 2.66 x 10-22 m3• The

mass of 1 chain follows from the mol ar mass: 4.67 x 1 o- kg. Th is gives a density of one polymer coil of 1.75 kg/m3

.

In reality PE has a density of 855 kg/m3, so the polymer coils do nat lie next to

each other in the material, but are highly entangled. This is very important for the mechanica! properties, because the long chains can only move past each other with difficulty due to the entanglements.

lf we try to stretch a polymer chain completely, then the di stance between the chain ends can in theory be re = n5Ls. which leads to an elongation:

For the PE we used in the previous example, we then get: À = 38.6. So, a polymer can be stretched very much when the chains are uncoiled, as shown in Figure 6.14.

~ Shoulder ~------------------~

Random orientation of polymer molecules

in cold drawn strain-hardened areas

Figure 6.14. The effect of stretchinga polymerie material.

Stretch

In reality, À will be lower, since most bands cannot be stretched to be completely straight. A polymer chain with C-C bonds will e.g. have angles of e = 1 09.SO, and the maximum length will be: re = nsLs sin(9/2) = "_,nsLs. inslead of nsLs.

L o}Yr? >: ~

e Besides the entanglements between the polymer chains, there are of course also

the normal intermolecular interactions: always van der Waals farces, sametimes dipole-dipole interactions ar hydragen bands. These can also change the shape and behaviour of the coils.


6.2.5 Overview

Below, an overview is given of the various structural elements that we find in synthetic polymers on the micro and meso level.

Micro

Atactic lsotactic Syndiotactic

Branched

Meso

Partially crystalline Ligthly cross-linked Close network

95

96 Matefiats Science tor Biomedical Engineering

6.3 Biopolymers

We wiltlook at proteins, potysaccharides and poly(hydroxyalkanoates). All are built up from optically pure monomars (amino acids, sugars, and hydroxy-acids). Biopolymers usually have polydispersity D = 1, because they are constructed monomer by monomer. Of course, there can be different lengths for different purposes. Also, there are usually no random copolymers: the sequence is determined by the genetic program, executed by enzymes.

6.3. 1 Proteins

Proteins are polymers of amino acids via amide bands (the primary structure) that can fold in water to secondary structures (a-helix, ~-sheet; via amide H-bonds), and tertiary structures (determined by hydrophobic interactions, H-bonds and electrastatic interactions, solidified by S-S bridges). In fact, proteins are polyamides like nylons. Because there are various side groups present (so proteins are copolymers), some people also speak of "decorated nylon".

As far as structural proteins are concerned, we do not find the folded globular structures as in enzymes, but fibrillar structures. Often these fibres are crosslinked to farm elastic networks. The best known examples of structural proteins are collagen, keratin, and spider dragline silk as typical strong fibres, and elastin as typical rubber.

Collagen is the most common protein fibre. There is a primary structure of ca. 1000 amino acids, where the triad Gly-X-Pro/Hyp is repeated. Hydroxyproline (Hyp) is a special farm of praline with an OH-group added. This chain forms a 280-nm long left-handed helix. Three of these chains farm a right-handed superhelix, and are kept tagether by hydragen bands and covalent crosslinks. This is called tropocollagen, with a molar mass of 285 kDa, a lengthof 300 nm, and a diameter of 1-5 nm.

Because there are several variaties of proteins that can be combined in the triple helix, we find different types of collé:~gen. For example: type I is the most camman one, in stretched tissues such as tendon, skin and bones; type 11 is found in connective tissues that are compressed, such as cartilage; type 111 is found in skin and blood vessels.

The further build-up follows a hierarchical pattem (see Figure 6.15): multiple tropocollagens combine in a microfibril, and these combine again to a subfibriL These farm fibrils, which are grouped tagether in fascicles, and finally a tendon is formed.

tropocollagen 1.5 nm

subfibril 10-20 nm

microfibril 3.5nm

faseiele 50-300 11m

tendon 100-500 11m

Figure 6.15. Hierarchical build-up of collagen in tendons.

6 Organic Materials I 97

Keratins are found in hair (also wool), nails and skin, and in animals also in horn, teathers and hoofs. The proteins have various amino-acid sequences, but there are always many S-S bridges. In the relaxed state there are many a-helices, but upon stretching these unfold to [3-sheets.

Here also we find a hierarchical structure: the a-helices all have a repeating sequence of 7 amino acids (a to g), of which 2 are hydrophobic in nature (a and d). In a top view (see Figure 6.16), this works out as follows (from a to g the helix runs into the paper): the apolar amino acids a and d end up precisely next to each other on one side of the helix.

g e'

r

Figure 6.16. Top view ofthe a-helices in keratin.

This leadstoa hydrophobic strip on the helix (see Figure 6.17), and two helices in water will now associate by attaching these strips to each other. This gives a socalled coiled-coil structure: a left-handed superhelix.

Figure 6.17. Two a-helices combinetoa superhelix by hydrophobic interactions.

Two coiled coils now combine to a protofilament, and subsequently two protofilaments combine to a protofibril. When a total of 4 protofibrils coil around each other in a right-handed fashion, this is called a filament. T ogether with a matrix, this then forms a composite material: the microfibriL Many microfibrils form a macrofibril, which then cluster logether to form a hair, in a similar hierarchical build-up as seen for tendon.


Silk consists of the protein fibroin, which forrns J3-sheets via hydragen bands. The amino acid sequence has as repeating unit the hexapeptide Ala-Giy-Aia-Giy-SerGiy, in whieh a smatt and a large amino-aeid: side ctu:rin alternate. Because of this structure, two fibroin chains can fit tagether like a zipper, as seen in Figure 6.18.

0.35 nm

Figure 6.18. {3-Sheets offibroin where the sma/1 amino acids (G/y) and the large amino acids (A/a or Ser) fit together.

The protein is semi-crystalline, so there are small crystalline areas in an amorphous matrix. With this material, the dragline of a spider web is constructed. lt is very important to realize that the silk fibres are spun by the spider, which directly gives a preterred orientation to the fibres. This trick can now be duplicated by chemists for synthetic polymers.

Elastin is a protein that is the typical bio-rubber: it is very elastic. lt leads to tissues with a yellow colour, such as the elastic cartilage in the ear and blood vessel walls. During aging, calcium phosphate crystallizes on the elastin fibres, and the tissues become less elastic: e.g. skin starts to wrinkle.

Elastin consists for 55-70% of the repeating sequence Gly-Aia-Ser. Most of these amino acids (60%) are apolar, and many 13-turns occur. Therefore, helices are formed, which can unwind upon stretching. There are fibres with a diameter of 5.5 nm and a length of 5-7 ~-tm. The protein has many intermolecular crosslinks (particularly via lysine side groups), and therefore is a typical rubber (see Figure 6.19).

-Figure 6.19. E/astin can be reversibly stretched due to its many crosslinks. Left

shows the molecules in rest, right shows the stressed form.

6 Organic Matenals I 99

6.3.2 Polysaccharides

These are polyacetals, and they are found in many farms: in humans we know glycogen, and in plants there is starch, but these are both starage polymers for glucose. As structural material only cellulose (in plants) and chitin (in insects) are of importance. In humans, we find many modified sugars, which can form gels in water: the hyaluronic acids. These are of importance mainly in composita biomaterials (see Chapter 9).

Cellulose is a polymer of glucose via 1 ,4-bonds. lt is of importance that glucose only has the 13-configuration in this polymer: this allows many H-bonds in the chain, which increases the strength of the materiaL With a-glucose we get glycogen, which is not a good structural materiaL

HO~:Q HO~OH

OH

-H~H HO '0 H 0

OH O CHPH

Cellulose farms fibrils with a diameter of 3.5 nm from ca. 40 molecules via Hbonds between the chains, and then combines the fibrils to larger fibrils (20-25 nm diameter). In fibres, these fibrils are wound in a specific angle, which is very important for the stiffness of the materiaL

Gels of polysaccharides are formed when the sugars are modified with acid groups, such as sulphonic acids or carboxylic acids. In water these will form polyelectrolytes, which are highly swollen with water, and can form networks. In humans, the most important examples are the glycosaminoglycans (GAGs), long polymers that occur in five classes: - Hyaluronic acid: repeat of N-acetyi-13-D-glucosamine and 13-D-glucoronic acid. Occurs in skin, cornea, heart valves, and synovia! fluid.

HO ~ \O:ll HO~O--..--Û~O-:--J-T-ó

NH 8 ooc o=(

CH3

- Chondroitin-4-sulphate: repeat of N-acetyi-13-D-galactosamine-4-sulphate and 13-Dglucoronic acid. Occurs in cartilage, skin, and cornea.

100 Materia/s Science tor Biomedica/ Engineering

- Chondroitin-6-sulphate: repeat of N-acetyi-P-D-galactosamine-6-sulphate and p-Dglucoronic acid. Occurs in cartilage, tendon, and intervertrebral discs.

- Dermatan sulphate: repeat of N-acetyi-P-D-galactosamine-4-sulphate and p-Liduronic acid. Occurs in skin, tendon, ligaments, and heart valves.

- Keratan sulphate: repeat of N-acetyl-p-D-glucosamine-6-sulphate and p-Ogalactose. Occurs in cartilage, cornea, and intervertrebral discs.

6.3.3 Poly(hydroxya/kanoate)s (PHAs)

These are isotactic polymers made by bacteria, which use them for energy storage. They can farm up to 80% of the dry cell mass, and are therefore quite easy to isolate. The most camman one is poly(D-3-hydroxybutyrate) (PHB).

-(0-CHR-CHrCO)nPHB: R = CH3

PHB is rather stiff and brittle, but this can be improved by adding propanoic acid to the nourishment of the bacteria. Th is leads to the formation of a copolymer of hydroxybutanoic acid (hydroxybutyrate) and hydroxypentanoic acid (hydroxyvalerate) with better mechanica! properties.

6 Organic Materia/s I 101

Problems

6.1. What is the difference between configurations and conformations in polymer chains?

6.2. What is the degree of polymerization of a poly(styrene) sample that has a number-averaged molar mass of 129 kg/mol?

6.3. lf the degree of polymerization of poly(vinyl chloride) is 729, what is then the molar mass of the polymer?

6.4. A copolymer of vinyl chloride and vinyl acetate (C4H502) contains a ratio of 19 partsof the farmer to 1 part of the latter. lf the number average molar mass is 21 kg/mol, what is the degree of polymerization?

6.5. A certain rubber is composed of equal weights of isoprene (C4HsCH3) and butadiene (C4H5). a. What is the male fraction of each of the rubber components? b. How many grams of sulphur must be added to 2 kg of this rubber to crosslink 1%

of all the repeating units? Note: 1 S atom crosslinks 2 repeating units.

6.6. A PP polymer contains equal numbers of macromolecules with 450, 500, 550, 600, 650, and 700 repeating units. What is the average molar mass by number and by weight?

6.7. Below, molar mass datafora poly(propene) material are tabulated. Compute: a. The number-averaged molar mass. b. The weight-averaged molar mass. c. The number-averaged degree of polymerization. d. The weight-averaged degree of polymerization. e. The polydispersity.

Molar mass male ma ss [kg/mol] fraction fraction

8-16 0.05 0.02 16-24 0.16 0.10 24-32 0.24 0.20 32-40 0.28 0.30 40-48 0.20 0.27 48-56 0.07 0.11


6.8. Poly(ethylene) (PE) has a density of 1000 kg/m3 in crystalline farm, and a density of 865 kg/m3 in the amorphous state. a. When a ünear PE ha8 a density of 970 kg/m3

, what is then the mass fraction of crystalline polymer? Hint: Start with 1 kg of polymer, and split it into crystalline and amorphous parts.

eall the crystal mass fraction x, and proceed from there. b. A branched PE has a density of 917 kg/m3

. What is the mass fraction of crystalline polymer in this material?

c. Why is the density of branched PE lower than that of linear PE?

6.9. In crystalline forma polymer melts at 240 oe, but in amorphous form it has a glass transition at 90 oe. The coefficient of thermal expansion of the liquid polymer is three times that of the solid polymer. lf the polymer volume expands linearly with the temperature in both cases, what is the ratio of the free volume at 100 oe and at 200 oe?

6.10. Give the reason why the glass transition temperature (Tg) differs for the following pairs of polymers with similar chemica! composition and the same molar ma ss: a. PE (150 K) vs. PP (250 K) b. Poly(1-butene) (249 K) vs. poly(2-butene) (200 K) c. PEO (232 K) vs. poly(vinyl alcohol) (358 K) d. Poly(ethyl acrylate) (249 K) vs. PMMA (378 K)

6.11. A polypeptide can sametimes be treated as a random coil. lf the length of an amino acid is 0.34 nm, what are then the average radius and the radius of gyration of a 1 00-amino-acid polypeptide?

6.12.* We have a polymer with Mn = 9000 g/mol, and we wish to lower the Tg by lowering the average molar mass. For this polymer it is known that: Tg,oo = 164 oe and CT= 98866 oe·g/mol. How many moles of the same polymer with Mn = 1 00 g/mol must be added to 1 male of the original polymer to lower its T9 to 121 oe?

7 Organic

Materials 11:

Mechanica I properties of (bio )polymers

In this chapter we will apply the structural knowledge of organic materials to understanding their mechanica! properties. Since these are quite different from the ones seen for inorganic materia Is, we will first give an overview of what to expect, and then treat each property in detail.

Compared to the inorganic materials, we now have very weak bands between the molecules. That means that elastic and plastic deformation occur simultaneously: bands are stretched so easily, that bond breaking and viseaus flow (as for glasses) occur together. That is why we speak of visco-elastic behaviour.

Nothing is too wonderful to be true if it be consistent with the laws of nature, and in such things as these, experiment is the best test of such consistency.

Michael Faraday (1791-1867)


7.1 Overview

The machanical properties of polymers are highly dependent on the temperature: polymers are visco-elastic, and can behave like glasses, rubbers (elastic deformation), or flow like viseaus liquids. Th is viseaus part absorbs energy, so the cr,Ecurves always show hysteresis (see Chapter 2).

Also, the strength and toughness are different for the glass and rubbery state: below the glass transition (Tg) there is brittie fracture like in ceramics, while above the Tg there is yield and plastic deformation as in metals (see Figure 7.1 ). The yield stress is therefore strongly dependent on temperature and the speed of the measurement (faster gives highervalues for crv).

80,---------------------------------------.

40 oe

68 oe

122oe

140 oe

0 0.10 0.20 0.30

Strain

Figure 7.1. Mechanica/ properties of polymers are dependent on the temperature.

lt is also very important always to consider what the structure of the polymer is: we can have highly crystalline materials, glasses, semi-crystalline materials and rubbers (elastomers). These all have different mechanica! properties, as shown in Figure 7.2.

7 Organic Materials IJ

Polymer singlecrystal fibre

Strain

Semi crystalline polymer

Rubber

Figure 7.2. Mechanica/ properties for different types of polymers.

As a general overview, we can use Table 7.1 in which the different types of machanical behaviour are related to the crosslink density.

Table 7.1. Possible mechanica/ behaviour for polymers.

Glass/ceramic Visco-elastic Rubber elasticity

Thermoplast T< Tg T> Tg -High crystallinity Low crystallinity

Elastomer T< T9 - T> Tg

Thermoharder Always - -

Intermezzo

105

The viscosity of polymers is obvious at low molar masses: they appear as molasses. At higher molar masses, the viscosity bacomes so high that the polymer appears as a solid. However, upon increasing the temperature, the viscosity will decrease again, following the so-called WLF-equation:

17.4(T g) log(ll) log(llo) 51.6+(T-Tg)

Here, 11o is 1012 Pa·s, and T9 is the glass transition temperature of the materia I.


7.2 Stiffness

The Young's modulus will bedependenton the temperature for most polymers, due to the glass transition. Of course, the amounts of crystallinity and crosslinking are also important, as can beseen in Figure 7.3.

glassy 1 00% crystalline glassy

:92 Q)

~

I~h·~ ~

"' "' Cl Cl g g "' "' :::1 :::1 ::; s "C "8 0 E E 0

~ i$ rubberv co co äi

amorphous äi 0 8 0

"' viseaus "' no crosslinking ·:;; ·;;; viseaus

Tg temperature Tm Tg temperature Tm

Figure 7.3. Young's modulus vs. crystallinity (left) and crosslinking (right).

Below T9 a polymer is glassy, and the Young's modulus depends on the strengthof the bands between the chains, as shown in Figure 7.4.

~ E z Q. lU

103 ---------------------------------Diamond

1Q-3

Epoxy Nylon Alkyds

10·2 1Q-1

Heavily crosslinked polymers

Covalent cross-link density

,.__.All covalent

Figure 7.4. The crosslinks and the temperafure delermine the modulus.

7 Organic Materlafs 11 107

7.2.1 Polymer crystals

Polymer crystals should behave as expected for inorganic crystalline materials, but this is not true: there is a very strong anisotropy in the Young's modulus. In the plane of the lamellae (see Figure 7.5) there are moduli of ca. 1 GPa (weak interactions between the chains), while perpendicular to the lamellae the moduli are ca. 100 GPa (the chains must be stretched).

Figure 7.5. Lamellar structure of a polymer crystal.

The latter modulus can be predieled from the farces that are necessary to change bond angles and bond lengths in polymer chains.

In contrast to inorganic materials, polymers can never reach theoretica! stiffness va lues (although we can come quite near), since it is nearly impossible to get 1 00% crystalline materials without defects.

Examp/es: - Poly(propene) theoretically has a modulus E = 50 GPa, but in bulk PP one finds only E = 2 GPa, and for PP fibres E = 20 GPa. -A specific nylon (PA66) even has E = 160 GPa in theory, but in the bulk only E = 3.2 GPa is found, and in fibres E = 5 GPa.

7.2.2 Seml-crystalllne po/ymers

When there are both crystalline and amorphous parts in a polymerie material, the amorphous part gives flexibility, while the crystalline part gives the material strength. For crystallinities up to ca. 20%, there are smal! crystals in an amorphous matrix, and these polymers behave like crosslinked rubbers. When the crystallinity rises over 40%, the crystals start touching each other, and the material becomes stiff, and less visco-elastic.

In polymer crystals, the lamellae must unfold upon deformation (see Figure 7.6). This absorbs a lot of energy, and therefore semi-crystalline polymers have a high toughness (high Wf), but are nat very stiff (E = 2 GPa). First, the amorphous between the lamellae is stretched, and then the lamellae themselves are reoriented. Then, the crystalline blocks are torn apart, and the segments are oriented with the amorphous chains in the draw direction.


Figure 7.6. The various stages of deformation fora semi-crystalline polymer: stretching of amorphous part, deformation of crystals, and breakup of crystals.

7.2.3 Glossy state

The Young's modulus of the glassy state (amorphous, below T9) is determined by both the primary, covalent interactions in the chain and the secondary, intermolecular farces between the chains:

1 f 1-f -= +-E EP Es

Dependent on the contribution of Ep (value of f) the modulus can be as high as 100 GPa for oriented nylon fibres. This is camparabie to e.g. copper.

7 Organic Materials 11 109

7.3 Visco-elastic behaviour

In order to describe the visco-elastic behaviour of polymers in a mathematica! way, one usually takes a mechanica! model. However, we can also understand it at the molecular level: the long macromolecules can move along each other due to the weak interactions between them, and in this way the material can easily be deformed in addition to the stretching of bonds.

7.3.1 Models

An elastic solid behaves according to Hooke's law, where the Young's modulusEis important:

a= Es.

This can also be written as:

da = E ds. dt dt

An ideal (Newtonian) liquid foilows a different law, where the viscosity TJ determines which flow speed is caused by a certain stress (or pressure):

ds. (}" =TJ

dt

A visco-elastic material has a mixture of these two properties, so it will show viscous flow due to stress, beside the elastic deformation. Th is can be seèn in two phenomena: (i) a constant stress willlead to permanent elangation (creep); (ii) in a polymer that is kept at a constant elangation the stress will diminish (stress relaxation).

In Figure 7.7, we can see what the response is toa certain action. On the left we see creep due to constant stress, and on the right a constant strain leads to stress relaxation.

constant stress constant elangation

creep

Figure 7.7. Greep (left) and stress relaxation (right) are typical for visco-elastic materials.


In order to describe these two types of behaviour, there are saveral models that use a mechanica! analogue: a spring for the elastic part, and a piston for the viscous part. We wiff look at the modelsof Maxwe#, Voigt (or Voigt-Ketvin), and linear eJastic solid (or Zener) model (see Figure 7.8).

E

Figure 7.8. The three models used for visco-elastic behaviour: Maxwell, Voigt, and linear elastic solid.

7.3.2 Maxwell model

This simple model can (partly) describe stress relaxation: a spring (a1 = Es1) and a piston (a2 = 11ds2/dt) are used in series, so the stress is the same:

The strains, however, must be added:

So for the strain rate we can write:

ds ds1 ds2 1 da a -=-+-=--+-dt dt dt E dt 11

Stress relaxation occurs when s is constant (se). so dddt = 0:

da E -=--a dt 11

With t, = 11/E, the relaxation time, and starting from t = 0, this integrates to:

Here we see that the stress indeed decreasas in time (see Figure 7.9). However, in real polymers the stress will never become completely zero as this model predicts.

7 Organic Matenals 11 111

(I)

~ &

i e " ~ 0

~ §

@ .g 8 &

Time Time

Figure 7.9. Stress relaxation (/eft) and creep (right) in the Maxwell model.

Creep is not predieled in a correct way at all by this model. Then we should have a constant cr (crc), and so dcr/dt = 0:

de= crc dt 11

Starting trom t = 0, this integrates to:

Th is is pure Newtonian flow, as fora liquid: the strain rate is constant, and does not level off (see Figure 7.9). A real polymer will have a maximum deformation tor a certain stress.

7.3.3 Voigt model

This model can (partly) describe creep. Here, the spring (cr1 = Ee1) and the piston (cr2 = 11de2/dt) are used in parallel, so the strains are the same:

Now, the stresses must be added, since bath elements contribute:

This gives forthestrain rate (with the same relaxation time •r = 11/E):

de cr e -=---dt 11 •r

By inlegrating trom t = 0, creep can now be described fora constant stress crc:


This is in good agreement with reality: creep is a non-linear process that reaches a maximum (ac/E), as shown in Figure 7.10. The deformation is completely reversible: when the stress is removed, the strain goes back to zero. However, there is no elastic behaviour at all.

~ c: 0 0 u. ~ ê .E .,

c: 0

~ ., ê e .E 0 ., u. 0

Time Time

Figure 7.10. Greep (left) and stress relaxation (right) in the Voigt model.

Stress relaxation is described incorrectly. When s is constant (se). and ds/dt = 0, it follows by inlegrating from t = 0:

a= Esc

This is purely elastic behaviour: a is constant (see Figure 7.10).

7.3.4 Linear etostic solld

Bath previous rnadeis each describe a part of the visco-elastic behaviour, but another part incorrectly. We can combine them in the linear elastic solid model. Here, we have a Maxwell-element (spring 1 +piston 2 in series) in parallel with a second spring (3). For such an assembly the following relations hold:

S = 1>1 + E2 = S3

ds1 a1 =TJ1dt=a2

a3 = E3E3

Aftera lot of mathematica! crunching, we get two relaxation times:

N.B.: These were called 'te and 'ta, respectively, in the first edition of this book.

7 Organic Materia/s 11 113

lt then follows that:

For creep (a constant at value crc from t = 0) this model then gives:

There is a non-linear change of the strain in timetoa maximum value (crc/E3), and there is a lso an elastic part of the deformation ( 'tRGc /'tKE3), which responds instantly (t = 0), as shown in Figure 7 .11.

Time Time

Figure 7.11. Greep (/eft) and stress relaxation (right) in the linear-e/astic-solid model.

For stress relaxation (constantE= ec from t = 0) we can write:

There is now a relaxation to a non-zero value (E3Ec), as seen in Figure 7 .11. So, this model can adequately describe bath creep and stress relaxation.

Remember, however, that it is just a mechanica! analogue: we did not put the molecular structure into the description anywhere.


7.3.5 Dynamic mechanica/ measurements

In more realistic tests, the polymer is deformed in a dynamic fashion: it is stretched and relaxed periodically, either by tension or by rotation. This is the case for e.g. joints during walking, heart valves, and veins. In these cases, we can say that the deformation follows a periadie sine function:

e = emax sin( rot)

Here, we use the angular frequency ro (rad/s), which is related to the normal frequency (v [Hz]) by ro = 2nv. The stress in the material will usually follow the strain (elastic behaviour), but there will be a time lag due to the viscous flow that also occurs:

(J = <Jmax sin( rot+ o)

The angle o is the so-called phase angle, which indicates how much difficulties the stress has to follow the applied strain (see Figure 7 .12).

ö/ro

c

~ .... 0 (/)

~~~--~--~---+~~---r~----.--r------~~

U5

21t/ro

Figure 7.12. In a dynamic test of a visco-e/astic material, stress and strain do not a/ways follow each other directly.

For an ideal elastic material we see that o = 0, since the stress directly follows the strain. In a visco-elastic material the stress cannot follow the strain and ö is not zero. lf we define E = crmaxlemax. we can write tor the stress:

(J = SmaxE sin( rot+ ö)

According to the usual goniometrie rules one can write:

a= emaxE cos(8) sin(mt) + SmaxE sin(8) cos( rot)= Smax{E' sin(mt) + E" cos(mt)}

7 Organic Materials 11

Here, 8 is defined via the relation:

E" tan(8)=-p

and we have defined:

E' = Ecos(ö) and E" = Esin(ö)

Some people prefer to write these two components of E as a complex number: E* = E' +iE". lt canthen be treated as a single quantiJ?'. There is then also a magnitude of this complex number: IEl = (E' 2 + E" 2)

11 •

115

For a perfeetry elastic material the stress is the same sine tunetion as the strain (in-phase behaviour), so 8 = 0, E' = E and E" = 0. Because of this, E' is called the elastic or storage component of the modulus (reversible storage of elastic energy). Fora liquid we have seen that cr = T]de/dt, so when e is a sine function, cr will follow a eosine tunetion (out-of-phase behaviour): 8 = 90°, E' = 0, and E" = E. Therefore E" is called the loss term: irreversible lossof energy due to viscous flow. You can actually calculate this energy loss during one cycle of the sine wave as the area under the cr,e-curve:

2rc/ro

W = J crde = ne~axE" [J/mÎ 0

The term tan(ö) can be seen as a measure for the internar friction due to flow, which leads to hysteresis in the cr,e-curve during stretching and relaxation.

7.3.6 Effect of trequency and temperafure

At very low frequencies, the polymer behaves like an elastic rubber: E' is small, and so are E" and tan(ö). The material is easy to deform, and there is little loss.

At very high frequencies, the polymer can no longer follow the deformation, and behaves like an elastic glass: E' is large, and E" and tan(8) are small. The material is hard to deform, but there is little loss.

In an intermediale range of frequencies, the polymer is visco-elastic: E" is large, and so is tan(ö). The material has much energy loss due to internar friction.

This behaviour is a lot like the changes due to temperature: high frequency is similar to low temperature (glass), and low frequency is like high temperature (rubber).

The preceding story can also be told with the shear stress and strain, which usually can be measured more easily (torsional movements). In that case, we can ju st replace E' and E" by G' and G".

During a variation of the temperature in a dynamic test, one can clearly see changes in the material due to e.g. the glass transition: they give a peak in tan(ö). This is why a technique like DMTA (dynamic mechanica! thermal analysis) is used a lot.


Example: The starage shear modulus and tan(ö) for atactic (amorphous) poly(styrene) as tunetion of temperature. a is the glass transition (movement of the main chain begins), and ~. y andeSare other transitions that are due to movementsof other part of the macromolecule.

,_.._ 'l'

9

E z 8 ::-

~ Ol 7

..Q

6

5

0 ;:o c: 19

-1 o; ..Q

-2 y

-200 -100 0 100 200

Temperature/oc

Can the models introduced above also predict this dynamic behaviour? The Maxwell model gives with E = Emax sin( rot):

E 1 tan(ö)=-=-

TtW ffi1r

Th is is a continuous decrease of the friction with frequency, which is nat found in reality.

The Voigt-Kelvin-model gives the opposite behaviour:

E'=E E" = 11ro tan(ö) = ; = ro1r

Here, we find a continuous increase of friction with frequency: also incorrect. The more realistic linear elastic solid model gives:

7 Organic Materials IJ

Here, tan(ö) has a maximum at an angular frequency:

1 ffi=

(see Figure 7.13), and the modulus increases with frequency. This is a good description of real polymer behaviour.

Elastic modulus IGI

!

1

lnternal friction tan o

Figure 7.13. The linear elastic solid model gives a good description for the mechanica/ behaviour of a visco-e/astic material.

For a biologica! material it is aften seen that tan(ö) has a flat maximum over a braad frequency range. This can be simuialed by putting many linear elastic solid modelsin series, as shown in Figure 7.14.

tan(&)

I~ logv

Figure 7.14. Model simu/ation for the behaviour of biologica/ materials.

117


Intennez.zo There is a better model for the behaviour of polymers than with springs and pistons, but it is too complicated to treat here. The basic ldea is that macromolecules move only along their length, as if they are restricted in a tube formed by the other macromolecu les. Th is is why it is called the "reptation" model. By describing the macromolecule as strings of beads conneeled by springs, located on a three-dimensional lattice, this model gives an expression for the characteristic time ('td) a molecule needs to diffuse out of its tube. With these times, the response of molecules to a dynamic force can be calculated, and it turns out the predietien for G' and G" as a function of frequency is quite good. However, because both the conformation and the movement of the macromolecules have to be described by statistica! models, the rnathematics are too complex to treat here.


7.4 Rubber elasflcity

Rubbers are elastic materials, but they display a very strange (non-linear) relation between stress and strain. This is due to the fact that the elasticity is not due to the stretching of bonds (change in energy), but to changes in entropy during deformation. A random coil is the most probable situation. (high entropy), while a completely stretched chain is very improbable (low entropy). Since entropy tends to a maximum, a force is needed to stretch a chain, even when there are no interactions keeping the chain together. Therefore, rubbers are often referred toasentropie springs.

In AppendixHit is derived (with the assumption L1V = 0) that the entropy change upon an elongation (or compression) A. follows the relation:

2 2 AS oc 3---A. À

From this it follows that AS < 0 for both stretching (A. > 1) and compression (A. < 1 ). Also in Appendix H, the following relation between stress and elongation is derived for a rubber with a shear modulus G:

This equation indeed describes the behaviour of rubbers quite nicely forA. between 0.4 and 1.3 (see Figure 7.15).

5

4

"' E 3 z

::;; 0 Û) (/)

2 I!? (i)

-1

-2 Û"'-----::-----:----,-------=--

2 3 4 5 6 7

À -3

Figure 7.15. Experimental and theoretica/ (dashed line) behaviour of rubbers during tension and compression.

The value of G can be related to the distance between the few covalent crosslinks in the rubber network, and therefore gives an idea of the possible holes in the network (indicated by the so-called mesh-size), in which small moleculescan be taken up. When the crosslink density increases, G becomes larger.


Also, we can write for the elangation factor À:

L À=-= 1 + e

La

For small values of e (e << 1) we can then approximately write:

Then the relation between a and a becomes:

a= G(1 + e- 1 + 2e) = 3Ge

So: G = E. Th is is exactly what is expected fora rubber with a Poisson ratio (v) of 0.5 3

(and indeed: in Appendix H, constant volume was assumed).

Intermezzo In addition to the unexpected machanical behaviour, there are two thermomechanical effects for rubbers that are also due to the importance of entropy changes.

1. The forceFin these materials is almast completely determined by the entropy (see Appendix H}, so when the temperature changes:

aF aT

as -->0 eL

This means that when the strain is constant, the force increases when the temperature rises: rubbers want to shrink at higher temperatures, in contrast to normal materials.

2. lt can also be derived what happens with the temperature when you stretch a rubber very quickly (so no heat can be exchanged with the surroundings):

• ~~ == -~~I:~ == :~I i > o

I So, rubbers will become warmer when you stretch !hem qu_ic_k~ly,_. _______ ___J

7 Organic Materlafs 11 121

7.5 Fracture

For fracture of crystalline or glassy polymers, Griffith's crack theory can be used as for inorganic materials. However, since the polymer chains can usually move past each other quite easily, we get the sameeffect as dislocation movement in metals: there is yield, and the work of fracture (200-2000 J/m2

) is much higher than the surface energy (1-1 0 J/m2

), so polymers are much tougher than ceramics. The fracture stress is dependent on the molar mass of the polymer, since for

longer chains there are more entanglements between the chains that prevent easy fracture growth:

We see that the tensile stress reaches a maximum value for infinitely high molar mass (crT,<X>). This is the same kind of behaviour we saw for the glass transition temperature in Chapter 6.

I nte111U!.ZZo Besides yield with necking, there can also be plastic deformation in polymerie materials by crazing: some of the chains are stretched, and this region appears quite different under the microscope. erazes look like cracks, but they are not.


7.6 Mechanica! properties of biopolymers

The mechanica! properties of biopolymers are aften strongly dependent on the surrounding medium: especially the water content is very important. Water acts as a plasticizer: it increases the mobility of the polymer chains. A common problem with studying these materialsis that in vivo and in vitro results are quite different: the remark "dead and fresh" is often seen. Keeping a biopolymer out of the body for a longer time, treating it against decay, or even drying it can drastically change the properties.

A general proparty of biopolymers is the surprisingly high fracture strength and toughness. This can be explained by the fact that most biopolymer materials consist of smaller modules, which are hierarchically built up to larger structures. Since the modules can be stretched by unfolding them one by one, the material as a whole can absorb a lot of energy before it is stretched to breaking (see Figure 7.16).

long molecule

long molecule with modules

z c:

ê 0

LL

1.0

short molecule

long molecule with module

long molecule

0.0 1.!::1.."===:::::::::::::::.__ ___ J._

0 50 100

Extension/nm

Figure 7. 16. Mechanica/ behaviour for short, long, and modular materials.

7.6.1 Proteins

Co/lagen: in dry form it is rather stiff and brittie (E = 6 GPa). When wet it is softer: first the modulus E is very low, because the folded structures in the fascieles are unfolding (the "toe" in Figure 7 .17), and above a certain strain a modulus of E = 1-1.5 GPa is found (the "heel"). The maximum elastic strain is 1-2%. Collagen in vivo is often continuously under stress by movement (e.g. tendons) or prestressing (e.g. in walls of blood vessels). At a maximum strain of 10% collagen breaks (fracture stress: 10-100 MPa).

toe

strain

Figure 7.17. General mechanica/ behaviour of co/lagen.


Keratin: Most materials with keratins are actually composites, since there is an amorphous matrix around fibres. Up to a strain of 2% keratin is elastic with E = 4 GPa. After that it shows yield: the modulus E bacomes much lower, since the ahelices unfold to p-sheets. At a strain of 30%, E increases again, because all helices are completely unfolded.

Also here, water functions as plasticizer: a higher water content gives yield at lower stresses and higher strains. The modulus of the matrix can decrease from 6 GPa to 1 GPa. In addition, these matenals become anisotropic at high water content, since the fibres start crienting in a parallel fashion in the ever softer matrix.

Keratin materials show a large hysteresis, and are therefore hard to break: the work offracture is 10 kJ/m2

, which is typical for modern high-performance composites. We wiltlook at the behaviour as composita materiallater in Chapter 9.

Example: Wool (sheep hair) shows a strongly different behaviour when the humidity in the air changes from 0% to 100%. Th is is the same reasen why hairdressers wet hair before cutting it.

300

'I' E 200 z ::2 (i) (/)

~ (i) 100

0 0 0.2 0.4 0.6

Strain

Silk: This is a semi-crystalline polymer, in which the fibres are strongly oriented due to the natura! spinning process. The crystalline areas give strength, while the amorphous matrix ensures elasticity. Again, the presence of water is very important.

Along the fibres silk is very stiff (E = 10 GPa), but due to the weak interactions between the layers it can stretch tostrains of 25-28% until it breaks at 1-1.3 GPa. The work of fracture is 12-18 MJ/m3 These properties are very similar to modern nylon fibres.

Elastin: Th is protein behaves like a typical rubber, with G = 0.6 MPa. Above a strain of 100% the stress increases strongly, since the crosslinks are now Ioaded (see Figure 7.18).

124 Materia/s Science tor Biomedical Engineering

25

20

z 15 :i!: Û! !/) Q) 10 ,_

èi5 5

0 0 0.5

Strain

Figure 7.18. Mechanica/ behaviour of elastin.

A good model forelastin is the protein poly(GVGVP), which shows contraction upon raising the temperature, or upon the addition of salts or acids. These latter effects are due to the decrease of electrastatic repulsion, since coo- is transformed to COOH (acid}, or by decreasing the dielectric constant (salts).

7.6.2 Polysaccharides

Cellulose fibres: Because of the many hydragen bands, cellulose is very stiff (E = 140 GPa), but this decreasas u pon the addition of water. The angle with which the fibres are wound in the material is also very important: E can become 8 times smaller when the angle changes from oo to 40°.

Gels: These are visco-elastic materials that are more on the viseaus side. There are only a few stabie crosslinks, so flow is very easy. Same plant gels (alginates) can be more elastic, since there are more crosslinks. Gels are usually parts of composita materials (see Chapter 9). Of the various type of GAG, hyaluronic acid is found most frequently, e.g. in the vitreous matter in the eye and in the synovia! fluid.

Due to the large quantities of water in these polyelectrolytes, the volume of these gels is very high and there is a lot of overlap between the chains, giving weak crosslinks. At high shear rates, hyaluronic acid gels change from elastic to more viseaus behaviour (shear thinning), which is used in joints to give good lubrication.

7 Organic Matenals 11 125

Problems

7.1. Give the expected machanical behaviour of thematenals given below. lndicate for each pair the most important difference between the materials. a. lsotactic linear PP (Mw = 120 kg/mol) vs. atactic linear PP (Mw = 100 kg/mol). b. Branched PVC, OP = 2000 vs. heavily crosslinked PVC, OP = 2000. c. Random copolymer of styrene and butadiene (Mn = 100 kg/mol, 10% of the sites

crosslinked, 20 °C} vs. random copolymer of styrene and butadiene (Mn = 120 kg/mol, 15% crosslinked, -85 °C).

d. Poly(isoprene) (100 kg/mol, 10% crosslinks) vs. poly(isoprene) (100 kg/mol, 20% crosslinks).

7 .2. A piece of polymer is stressed to a level of 28 MPa and its length is held constant. After 30 days the stress has fallen to 17 MPa. How long will it take for the stress to fall to 7 MPa if this is a Maxwell solid?

7.3. Ten minutes aftera stress of 7 MPa was applied toa polymer rod with an elastic modulus of 700 MPa, the strain was measured to be 0.005. a. What is the viscosity of this polymer if it behaves like a Voigt solid? b. After 1000 minutes, what strain can be expected?

7.4. A polymer that can bedescribed as a linear elastic solid shows creep. a. lf::: = 0.005 at t = 0, and s = 0.02 after waiting very long, what is then the value of

'tR hK for this polymer? b. Describe this creep, and the constant strain after a long time, in terms of

macromolecules. c. Th is polymer shows maximum internal friction at ro = 1 0 rad/s. Calculate 'tK and

tR, and the value of tan(o) at ro = 5, 10, and 15 rad/s.

7.5. An elastomeric material is stretched 20% under a tensile stress of 0.25 MPa. Calculate the stresses necessary for stretching this material 50%, and campressing it 20%.

7.6. A poly(styrene) component must not fail when a tensile stress of 1.25 MPa is applied. Delermine the maximum allowable surface crack length if the surface energy of PS is 0.50 J/m2

• Assume a modulus of elasticity of 3.0 GPa (glassy state).

7.7. Fora plate of a brittie polymer with thickness (t) 5 mm and width (b) 25 mm we find a tensile strengthof 85 MPa, when there is a notch on the side of the plate. For this kind of situation it is known that the following formula holds for the fracture toughness:

K1c = Fa 112/(bxt) x {1.99- 0.41(a/b) + 18.7(a/b)2 38.48(a/b)3 + 53.85(a/bt}

Here, F is the force on the materia I, and a is the length of the crack in the material. lf we know that K1c = 1.25 MPa·m 112

, calculate the length of the cracks in the material that was tested. Hint: Can this horrendous formula perhaps be approximated by sarnething simpler?


7.8. For PMMA it is found that the tensile strength depends on the rnalar mass: 1 07 MPa at 40 kg/mol, and 170 MP a at 60 kg/mol. Preetiet the tensite strength at a mofar mass of 30 kglrnol.

7.9.* For the visco-elastic behaviour of poly(styrene) it is known that: E = 3 GPa, tK

= 0.01 s and tR = 0.005 s. a. Calculate the energy that this polymer takes up during a dynamic laad of ro = 100

rad/s with a maximum strain of 0.02. b. Explain this energy uptake in terms of macromolecules. c. Why could this energy uptake be relevant for a biomedical application of this

polymer?

8 Changing

material

properties 1:

M icrostructu re

In order to change the material properties of existing materials, there are two quite different approaches:

(i) Try to change the structure of the material itself, by some processing technique, or by choosing the chemica! composition in a particular way. These possibilities will be treated in this Chapter.

(i i) Try to combine different types of materials in a composite. Th is is aften found in biologica! materials, and will be treated in the next Chapter ..

The only actlvities worth doing in the long run are the searching of knowledge and the creation of beauty.

Arthur C. Clarke (1917-2008)


8.1 Changing metal properties

In metals, the mechanica! properties (except elasticity!) are mainly determined by imperfectionsin the crystal structure. Specifically, the mobility of dislocations is important for the yield strength, and the growth of cracks delermines the fracture strength. This last proparty is also dependent on the workof fracture, or the fracture toughness, which itself is strongly dependent on the mobility of dislocations.

lf we want to change the mechanica I properties of these materia Is, we must therefore change the mobility of the dislocations. When the mobility of the dislocations is decreased, the metal gets a higher yield strength and a higher tensile strength (for ductile fracture), but its toughness will go down: stronger but more brittle. When the mobility of the dislocations is increased, the yield stress will go down, but the toughness will increase: weaker but more ductile. Toughness and strength are therefore contradictory demands!

N.B.: For metals we aften say strengthwhen we are actually discussing the yield stress, since this indicates the start of plastic deformation, which is detrimental for its use as construction materiaL However, sametimes the fracture stress ar ultimata tensile stress is meant. Be careful!

8.1.1 Strengthening

The mobility of dislocations can be decreased by creating more dislocations that interfere with each other, by creating higher lension in the crystallattice, ar by making more barriers for movement Methods for this are: - Materialsin which several phases (regions with different chemica! composition) are present. We canthen speak of a specific microstructure, with many barriers. This will be treated in more detail in section 8.2. - lnserting atoms with a different diameter in a lattice also ereales more tension. These materials are aften called solid solutions (see Figure 8.1 ). - Precipitation hardening gives more tension by inserting small crystals of another material in the lattice (see Figure 8.1 ).

ooooeoo 0000 000 oeo ooo 00000000 ooooooeo

0000 0 eoooo o

00000000

00000 w~'t:"-:!-+-+--+-+-+;-r-w 0

0 0 0 0

00 00000 0000

Figure 8.1. Solid solutions (left) and precipitates (right) lead to lower mobility for dis/ocations.

8 Changing Properties I 129

Examp/e: Cu/Zn alloys (brass ); yield strength increases when the Zn-content rises.

Solid-solution

I. (a) brass

0 30 {Pure Cu)

Weight% Zn

- Dispersion hardening is basically the same phenomenon as precipitation hardening, but with larger particles in the matrix. This will also give more grain boundaries, and therefore more barriers. Actually, we are now making a kind of composita of a metal and another material (see Chapter 9). - Cold working (or cold rolling, see Figure 8.2) creates more tension in the lattice by plastic deformation, and also increases the number of dislocations. Th is is the same as the strain or work hardening we mentioned in section 2.2.

Predeformation Postdeformation

Residual stress

Ductility Ductility Strength

Hardness Hardness

Recovery

lncreasing temperature after cold rolling ---+Figure 8.2. Gold working and annealing have a strong influence on

dislocation mobility.


8.1.2 Toughening

The mobiüty of disJocations can be increased by removing lension in the crystal lattice, or by removing barriers for movement Methods for this are: - Annealing (or tempering, see Figure 8.2) and recrystallization: this will restare the effects of cold working, and will lead to fewer dislocations and less tension. The toughness then goes up (as seen in the ductility or the reduction in area), but the strengthand hardness go down (see Figure 8.3).

1900~1 --···

CU a.. 1700

~ ..c:: - 1500 Cl c ~ "" (ij ~

1300 60 e.... "0 CU ]i Q) .... >- CU

"0 1100 50 .E c CU c ~ .Q ïi.i 900 40 ts c :::)

~ "0

~ 700 30

200 300 400 500 600 Tempering temperature/°C

Figure 8.3. Effect of annealing on mechanica/ properties of metals.

- Less contaminations in the metal: there will be less lension in the lattice. So a pure metal is easier to deform. - Materials with only one phase: fewer barriers due to phase boundaries. This will be discussed in the next section. - Changing the lattice type: FCC inslead of HCP/BCC gives an easier deformation (more slip systems). This again related to phase diagrams, and will be treated later.

Example: Iron can be forged better at higher temperature, since it then changes from a.-Fe (BCC) to y-Fe (FCC).

8 Changing Properties I

8.1.3 Graln size

The only parameter that can increase both the strength (yield stress) and the toughness is the grain size. This is true for both polycrystalline metals and polycrystalline ceramics.

131

The first effect has already been discussed: smal! grains have a lot of grain boundaries that restriet the mobility of dislocations, and therefore the yield stress is high. Thus, the material is stronger for smaller grains.

There is a general relation between the yield stress of a polycrystalline material (cry) and the grain size (d9), related to the yield strengthof a single crystal (crv,sc):

However, since smaller grains give more grain boundaries, and these are also weak places in the material, there are also implications for fracture. Cracks will preferentially grow along the grain boundaries. But for smal! grains, the grain boundaries forma very twisting path, and the cracks must change direction so often, that a lot of energy is lost before the crack is large enough to fracture the material. Therefore, there is a high workof fracture, and a high toughness.

Example: Flint is a form of microcrystalline quartz that is very hard. Transparent flint has a rupture strengthof 190 MPa, and a grain size of less than 1 J..lm. Opaque flint has much larger grains (2-10 J..tm), and alsoasmaller strength: 120 MPa.

An optima! and uniform grain size distribution is very important for the mechanica! properties of a polycrystalline materiaL Rapid cooling of a molten material often results in small grains. However, during cooling saveral other unexpected things can happen: for this, we have to take a look at phase diagrams in the next section.

132 Materlafs Science tor Biomedica/ Engineering

8.2 Phase diagrams

In order to understand a bit more about the effect of different phases in a material, and the effect on the microstructure, we must be able to read and interpret phase diagrams.

The definition of a phase is: a region of the material with a different aggregation state (gas, liquid, solid) and/or a different chemica! composition.

The most simpte example of a phase diagram is the case of one compound (so the chemica! composition is always the same) with different aggregation states. Figure 8.4 shows the phase diagram of water, where we only see which phase (here: aggregation state) is present at a particular temperature and pressure.

solid liquid

~ 1il 1 atm (/)

~ c.

0.006atm

vapour

0 0.0075 100

Temperaturerc

Figure 8.4. Phase diagram of water: depending on pressure and temperature, three phases can be present.

Much more important for materials science are the phase diagrams of at least two components: binary phase diagrams. Now the chemica! composition of the phases can also change. Usually we choose a constant pressure (1 bar), and look only at the effects of temperature and composition. Let's take a look at the phase diagram of capper and nickel (figure 8.5). Chemica! composition (always with symbol x) can be indicated in atomie percentages (mole fractions), but more commonly in weight (or mass) fractions. Here we will use wt% nickel.

ü

~ ::J

~ (IJ a. E Q) 1-


1500 1455 oe

Liquid (L)

1400

1300

1200

Solid (S)

Cu 20 40 60 80

Weîght percent nîckel

Figure 8.5. Binary phase diagram of copper and nicke/ at 1 bar.

What can we see in this binary phase diagram? At 0 wt% Ni (pure Cu) and 100 wt% Ni, we see the melting points of the pure compounds. When we take a melt with 65 wt% Ni (Co) at 1450 oe (a), we have one phase: a liquid. Therefore, this region of the diagram is marked "L". At low temperatures (c), we see only a solid (S), which is a mixture (an alloy) of capper and nickel. Of course, the chemica! composition hereis also 65 wt% Ni. However, during the cooling process, several strange things happen, and these can be read from the phase diagram.

When we cool the liquid starting trom 1450 oe, we will see sarnething happen for the first time at ca. 1360 oe, when we reach the area marked "S+L ". This area is a torbidden area for this mixture: it must separate into a liquid phase (on the liquidus line) and asolid phase (on the solidus line). What is very important is that the first solid material to come out of the liquid has a chemica! composition of ca. 78 wt% Ni: it contains much more nickel than the liquid. This means that Ni tends to solidify befare Cu, and that Cu will follow at lower temperatures.

At point (b) at ca. 1340 oe, this has changed: the solid (b') now has a composition of 72 wt% Ni (Cs), and the liquid (b") has a composition of 60 wt% Ni (CL). Since the liquid and the solid phase have the same temperature, we can conneet them with a horizontalline: the tie line.

When we cool even further, we will continue to see a change in composition of the phases: the solid phase will fellow the solidus line, and the liquid phase will follow the liquidus, both rnaving to the left.

At a temperature of ca. 1310 °C, we reach the area marked "S": this means that we will see one phase only, which is a solid. The composition of this solid will be (of course) 65 wt% Ni. After that, the solid will simply cool to e.g. 1100 oe (c).

Ni


During this cooling, we will see not only changes in the chemica/ composition of the liquid and the solid, but their quantities must also change: we start with 100% L, and we mustand up with 100% S. How can we see what happens in between? For this, we can use a simple construction with the horizontalfine that connects the solid and the liquid: it is cut into two parts by the average composition, which must of course stay the same in the whole experiment. We use the tie line to delermine the chemica! compositions of the solid phase (x1), the liquid phase (x2), and we know the average composition of the system (x0), which is also the starting composition. We canthen use the so-called lever rule to calculate the mass fractions of solid (f1) and liquid (f2) (the physical composition of the system):

(lf you want to be convineed why this rule works: see Appendix K.) Be careful: these~ arealso fractions (weight or male), but are e.g. wt% L, and

never wt% Ni! They reprasent the way the whole system is split into different phases.

To helpwithall the symbols, perhaps this simple example is of use:

mo= 5 Xo = 3/5

m1 = 2 '1 = 2/5 X1 = 1/2

m2 =3 '2 = 3/5 X2 = 2/3

So, we can catculate for every temperature during the cooling how much liquid and how much solid is present. And of course we can also see what the composition is of the phases that are formed at each tempersture (where the tie line intersacts the liquidus and the solidus). So, we can performa complete physical (amount of liquid and solid) and chemica! (composition of phases) analysis. Please take care: you must decide yourself which phases you call 1 and 2! lt is not always so that 1 is solid and 2 is liquid. lf you forget this, all your calculations will be the wrong way around.

In Figure 8.6 we see the cooling process fora copper-nickel mixture with 35 wt% Ni. The compositions of the formed phases are indicated. Use the lever rule to check whether the amounts of liquid and solid are correct. Note that the solid phase is now called o., sarnething that happens a lot in phase diagrams. You can always use the name as indicated in the diagram.

1300

1200

1100 20

Intermezzo

a

8 Changing Properties I

~ L(35Ni) ~

a.

30

I I I I I

0 I

wt%Ni

40

L

50

Figure 8.6. Cooling process fora 35 wfO!b Ni melt.

Why is this important tor the materials scientist? Since a solid material that is once formed will usually not change quickly in chemica! composition, we can somelimes see the formation of various layers during cooling, each with different chemica! composition. The first grains of solid will be 46 wt% Ni, and the last solid to form will be 35 wt% Ni. Since we saw earlier that the presence of strange atoms in a lattice will influence the mechanica! properties, this microstructure will be very important. The phenomenon of getting different layers in the grains is called coring, and will happen e.g. at rapid cooling trom the melt.

135


8.2.1 Phase rule

We can qonclude something else about the cooling process, but for that the phase rule must be introduced. We will notshow here why it is valid (see Appendix J), but we will simply apply it to our phase diagrams. The phase rule says that with a certain number of components ( C), and a certain number of phases (P), we can only choose a restricted number of parameters, the degrees of treedom (!f):

!F=C+2-P

What does this mean? We willlook only at binary phase diagrams: there will be two components (C = 2). The number of phases can be read trom the phase diagram: P = 1 for the regions marked "L~ and "S", and P = 2 for the region marked "S+L". We willlater see that P = 3 is also possible.

The parameters that can be chosen freely are usually: temperature, pressure, and chemica! composition. In our binary phase diagrams, we usually set the pressure at 1 bar, so that choice has been made: !Fis 1 lower than allowed by the phase rule. That leaves us with:

!f=3-P

So, when P = 1 (e.g. tor a liquid melt), we have !f = 2: we can choose the temparature and chemica! composition of the melt freely, as long as we stay within the region marked "L". When we are in the region marked "S+L" (both asolid and a liquid phase ), we have P = 2 and so !F = 1: we can set the temperature, but the chemica I compositions of both phases are completely determined by the solidus and liquidus lines. They are no longer free.

When we now look at a phase diagram like that of lead and tin (solder), we see another phenomenon: the eutectic (see Figure 8.7}.

350 327 ·c

300

L

250 232 'C u ~ :::> 200 ~ (X 183 ·c :g_ E

18.3 ~ 150 61.9 97.8

100 a.+(}

50

Pb 10 20 30 40 50 60 70 80 90 Sn

Weight percent tin

Figure 8.7. The phase diagram of lead and tin at 1 bar.

(.)

~ .a ~ ~ E (!) 1-

8 ehanging Properties I 137

Th is means that there is a possibility of three phases coexisting in the material: here at 183 oe we see that there are asolid (a, 18.3% Sn), a liquid (L, 61.9% Sn), and another solid ([3, 97.8% Sn) present at the same time. Now P = 3, so :r = 0: we can not choose anything freely. Bath temperature and all chemica! compositions are fixed by the diagram. This also means that the three-phase region is only a horizontalline, and not a two-dimensional region like the one-phase or two-phase areas. There can also be vertical lines in a diagram like this: compounds with a very narrow range for the chemica! composition. These can also occur on the y-axes!

Intermez.zo When cooling water, we have e = 1 (H20), and when the pressure is chosen at 1 bar, we have: :r = 2 - P. For liquid water, we have P = 1, so :r = 1: we can freely choose the temperafure (between 0 and 100 oe, of course). However, at 0 oe we also get ice (solid phase), and soP= 2, and :r = 0: we can withdraw heat, but the water and the ice will stay 0 oe until allliquid is gone, and we again have one phase (solid ice). Then we can lower the temperafure again.

Let's start by coolinga melt with exactly the eutectic composition of 61.9 wt% Sn (see Figure 8.8). There will be liquid until we reach the eutectic temperafure (183 °C). Then, both a and 13 are formed simultaneously as two new solid phases: we have 3 phases, so no degrees of freedom, and the temperafure stays constant. When we keep on removing heat, the liquid will finally disappear, and only a and J) will be left: we enter the "a+l3" region, and the temperature can start decreasing again. From the lever rule we can calculate that ju st below the eutectic there will be 55 wt% 13 and 45 wt%a.

350 327 oe

300 L, 61.9 wt% Sn

L' ' ' ' 250 ' 232 oe ' ' ' ' ' '

200

61.9 97.8 150

100 a+fl "--

fl: 97.8 wt% Sn

a: 18.3 wt% Sn 50

Pb 60 70 80 90 Sn

"' Weight percent tin

Figure 8.8. eooling a melt at 61.9 wt% Sn (eutectic composition) gives a layered structure of a (white) and f3 (dashed).


The solid that is now formed has-of course-the eutectic composition, but it also shows a characteristic layered structure. How is this microstructure formed? The only way in wh.ich both a. and p can ba formed simuJtaneously as solids from a liquid melt is by growing "sideways" in parallel layers, as shown in Figure 8.9. The atoms in the melt will move by ditfusion to the layers in which they preter to crystallize in higher concentration (Pb toa, and Sn to ]3).

growth direction of solid layers

Figure 8.9. The layered eutectic structure is formed by sideways growth.

What will happen during cooling of a melt with a non-euteetic composition? That depends on whether we are at lower levels of tin, or higher ones. We will first look at a level of tin below the eutectic composition, e.g. 40 wt% Sn.

350.------------------------------------------------------,

300

250

200

150

100

50

Pb

327°C l, ca. 61.9 l, 40::% Sn @__l

l 000

: o a ' '

a •

1~8.3 /~ ' : '

primary a 1

eutectic a eutectîc ~! 10 20 30 40

61.9

a+~

50 60

Weight percent tin

70

Figure 8.10. Cooling a meft of 40 wt% Sn.

l

232 oe

97.8

80 90 Sn


As can be seen in Figure 8.1 0, the melt will be one phase (L) until ca. 230 oe, where we reach the forbidden zone "o.+L". Here, the material must separate into a solid (a.) and a liquid (L): inthemelt there will be grains of a with a composition of ca. 17 wt% Sn. Further cooling will give more solid a., and the compositions of both a and L will change, as given by the solidus and liquidus, respectively. Just above the eutectic, the solid phase that forms will have a composition of just below 18.3 wt% Sn, and the remaining liquid will beat just below 61.9 wt% Sn. At this temperature we can use the lever rule, to find that we have 50 wt% a and 50 wt% L.

Just below the eutectic line, we have a microstructure of the solid material with particles of a in a layered matrix of o.+j}. All a phases (grains and layers) will have a composition of just below 18.3 wt% Sn, and all 13 layers have a composition of just above 97.8 wt% Sn. At this temperature we can use the lever rule to find a physical composition of 73 wt% a and 27 wt% J3.

Further cooling winlead to further composition changes: the a phases will contain less and less Sn, while the 13 phases will get more and more Sn. Also, the quantities of the phases will therefore change according to the lever rule.

But when we start with a composition above the eutectic composition, e.g. 70 wt% Sn, we first pass through the I3+L region befare we reach the eutectic. The final result is similar, except that we get particles of 13 in a eutectic matrix (try this yourselfl). This willlead to quite different machanical properties.

When we start at concentrations below 18.3 wt% Sn, we will never pass the eutectic line, and we will simply get formation of solid a. from the melt, which at lower temperatures will give some small quantity of 13 solid particles embedded in an amatrix, as shown in Figure 8.11.

350

@__ 300 (l . . .

250

~: . 9 : à> .... . a+L .a . 200 . ro • .... 183 oe (]) . a. ' E (l ' (]) ' 1- 150 ' t ~p, ca. 98 wt% S" . . . . .

100 . a+~

a, ca. 8 wt% Sn

50

Pb 10 20 30 40 50

Weight percent tin

Figure 8.11. Cooling a melt of 15 wt% Sn.


For very low Sn-concentrations (almost pure lead), we will not even see solid J3 appear, but we can get pure a from the melt, since we never reach the a+J3 region.

8.2.2 Steel

For a realistic, well researched, and important example, we willlook at the phase diagram of iron (Fe} and carbon (C}. For practical purposes, we usually display this as the phase diagram of the components Fe (pure) and the very stabie compound Fe3C (6.67 wt% C), since this almost always forms inslead of pure carbon (see Figure 8.12).

When the carbon content is tow enough (< 2.14 wt%), the material we get is called steel, and is indeed built up from Fe and Fe3C. For higher carbon contents (> 2.14 wt%), the material is called cast iron. Cast ironscan also contain pure carbon as component, but we wilt not discuss these any further here. Cast irons can be molten more easily, but are very brittle.

1538 ·c I L

1200 1147 ·c ~ Austenite, y ~

2.14 4.30

:::)

m 1000 ~ 912 ·c E (!!.

727 ·c

Ferrite,

600 0.022

Cementite, Fe3C

400--c

0 2 3 4 5

wt%C

Figure 8.12. Phase diagram of iron (Fe) and cementite (Fe3C).

In this phase diagram we see, besides the liquid melt (L), the foltowing solid phases: a-Fe (ferrite, BCC metal), y-Fe (austenite, FCC metal), &-Fe and Fe3C (cementite, brittie ceramic).

We wilt mainly look at the region of the diagram where austenite (y) is cooled.

6

This will in principle always give mixtures offerrite (a) and cementite (Fe3C), but the microstructure of the resulting material can be very different, si nee there is a eutectic line at 727 oe with the austenite composition at 0.76% C. Since territe is tough and

6.67

ü 0

~ ::>

~ Q) Cl. E Q) I-


ductile, but cementite is hard and brittle, this can dramatically change the mechanica! properties of the material.

Cooling austenite at exactly 0.76 wt% C (see Figure 8.13) will mean that other phases are formed for the first time on the eutectic lîne. There you simultaneously get a-Fe en Fe3C, which forma lamellar material: pearlite. Due to the overall composition, you get plates of cementite in an a-Fe matrix, since more a-Fe will be formed (check this wîth the lever rule). Also here, the formation of the lamellar structure occurs by a "sideways" growth. The austenite was formed as grains from the original melt, and the new phases will be formed at the graîn boundaries. Since ferrite is metallic, we get a ductile matrix with brittie ceramic plates in it.

a

1100

1000 ® y

900 i/ y + Fe3C

800 a+ y 727 oe

700 Fe3C

600 a+ Fe3C

500

400

0 2

wt%C

Figure 8.13. Coofing austenite at the eutectic composition, yiefding cementite (dashed) and ferrite (white).

For an austenite with a composition below 0.76 wt% C, cooling takesus through the "a+y" region (see Figure 8.14, left), so first a-Fe grains will be formed on the grain boundaries in the y-Fe. Th is a-Fe will therefore later farm the matrix of the fînal materiaL The last y-Fe particles will disappear and form a eutectic mixture at 727 oe: pearlite (layers of a-Fe and Fe3C). These pearlite particles will be embedded in a matrix of a-Fe. Since you now have a matrix of ductile ferrite, the material is not very strong (low crv), but quite tough.

ü

~ ~ Q) a. E a Q)

I-


1100

1000

900

800

700

600

0 2

wt%C

Figure 8.14. Cooling an austenite with less (left) and more (right) than 0. 76 wt% C.

Austenite with more than 0.76 wt% C will pass through the "y+Fe3C" region (see Figure 8.14, right), and first farm Fe3C grains on the grain boundaries in the y-Fe. So, the matrix of the final material will be cementite. The last y-Fe will disappear at the eutectic line, and we get pearlite particles in a cementite matrix. This gives a material with a very high yield stress, but it will break easily.

8.2.3 Kinetics*

Unfortunately, things do nat always happen the way it is described by the phase diagram. Changes trom one solid phase toanother solid phase (e.g. y ~a) are normally quite slow, so we do nat get the equilibrium structures.

One way of studying the effect of time is by keeping austenite at a certain fixed temperature below 727 oe, where it should change to territe and cementite, and see how long this takes. Wethen get a TTT-diagram (time-temperature-transformation), in which the solid lines indicate 1% and 99% transformation, and the dashed line is 50% transformation. Take care: the time scale in Figure 8.15 is logarithmic.

u ~ ~ :! E (!) 1-


800 A

700

Coarse pearlite

600

A F+C

500 Fine pearlite

400

300 Bainite

200

100

o' 1 min 1 h 1 day

0.1 10 102 103 104 105

Tirnets

Figure 8.15. TTT -diagram for the cooling of austenite (A), forming territe (F) and cementite (C).

We can see saveral things in such a diagram. One of them is that it takes a long time to change the phases at high temperature and low temperature, but only a short time at medium temperature. Why is this?

At high temperature, there is little driving force to form another phase (we are just below the eutectic), and we will slowly get a few new grains, which will then grow larger. The final material is coarse pearlite: it contains large grains.

At low temperature the driving force is large (far below eutectic), but the processas in these solids are quite slow (diffusion). Now, we will get many grains formed, but they will grow very slowly. The final material is called bainite, with many small grains.

In between, we have an optimum combination of driving force and growth speed: we will get a medium amount of grains with medium size. This material is called fine pearlite. The microstructure becomes finer (more narrow layers) when we go from coarse pearlite via fine pearlite to bainite. Th is means that the strength wil1 increase, but the toughness will decrease.

Another process occurs when we quickly cool (quench) austenite: we do not get pearlite at all, but martensite with a needle-shaped microstructure. This is formed without diffusion, which explains why it occurs so quickly. Martensite has a deformed crystal structure (body-centered tetragonal, BCT), which is actually not stable, but will slowly change into a cubic lattice (it is a metastable state). lt is very hard, because the carbon content is actually too high for this crystal structure, so we get a lot of tension. This internat tension can even be too large for the material to withstand, and

ü

~


we can get spontaneous fracture. Therefore martensite is often tempered (annealed), which releases some of the stress. We will then get the formation of cementite on the grain boun<faries, thus reducing the carbon content. The material will then be less strong, but much tougher.

In practica, the TTT-diagram is aften replaced by a CCT -diagram (continuouscooling-transformation), since continuous cooling at a certain speed is more realistic then annealing at constant temperature (Figure 8.16).

800 A

727

700

600 A

500

.a 400 ~

'

!i E (I)

1- 300

',,

-------,

200

100

', ...............

Pearlite + Fine Coarse 0 ~------M_a_rt_e_ns_it_e_L ______ M_a_rt_e_n_si_te __ ~ ______ p_e_a_rlit_e ______ ~ ____ P_ea_r_lit_e~

0.1 10 102 103 104 105

Time/s

Figure 8.16. CGT-diagram for the coo/ing of austenite. Ms indicates the startand MF the completion of martensite formation

The quicker you cool, the harder the steel that is formed: you can get almast completely martensite instead of pearlite or bainite. By adding small amounts of other metals, the change to pearlite can be slowed down further, and normal cooling will still give you martensite.

So, the general condusion is: although phase diagrams are very useful for predicting microstructures, the kinatics of the processing technique must not be forgotten.


8.3 Polymers

In polymerie materials, the mechanica! properties are strongly determined by the mobility and orientation of the macromolecu les. Both these factors are determined by the interactions between the macromolecules, and therefore by their chemica! structure. Since we can make copolymers, we can combine different chemica! structures in one macromolecule. This is necessary, since most polymers of different chemica! composition do not mix easily.

By now making block-copolymers, large blocks of different chemica! composition can be combined into one materiaL Often, the blocks of similar properties will cluster together (no mixing): we get a kind of phase separation on a very smal! scale (nanometers). This is often called microphase separation.

In addition, the different chemica! blocks in the macromolecules can form amorphous and crystalline parts, which can further increase the possibilities for combining mechanica! properties in a materiaL

Actually, you have now made a composita material, with e.g. hard blocks in a soft matrix. The volume fractions of both parts can be adjusted via the synthesis, and this willlead to different microstructures, and therefore to different mechanica! properties. Low amounts of one block lead to the formation of spherical phases of that block in a matrix of the other block. Longer blocks lead to cylinders in a matrix, and equal lengths of the blocks give a lamellar structure.

For block-copolymers we get a special behaviour at the surface of the material: the blocks of the macromolecule with the lowest surface energy (y) will be present much more on the surface. This means that the material presents a quite different surface to the environment than one would expect based on the average (bulk) composition. Of course, the same can happen for the v~rious additives that are usually added to polymerie materials in smal! quantities: they can aggregate at the surface, and have a much larger effect than expected.

Another possibility to change the mechanica! properties of a polymerie material is by stretching it: this leads to a more parallel orientation of the macromolecu les, which thereby get stronger interactions. This can lead to very strong fibres, since ultimately there are only covalent bonds in the draw direction.

ro a.. .:E

t c: !!! 'liî :2

~

146 Matenals Science tor Biomedica/ Engineering

Problems

8.1. A cyflndrlcaf specimen of cotd-worked Cu has a ductffity of 25 %EL lf it8 coldworked radius is 10 mm, what was its radius before deformation? Note: In the figures below, cold workingis indicated as: %CW = 100(Ao- Afinai)/A0

70 r--------------,

800

700

600

500

400 Bra ss

300

200

100

0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70

Percent cold work Percent cold work

8.2. A cylindrical rod of Cu, originally 16.0 mm in diameter, is to be cold-worked by drawing; the cross section will remain circular during deformation. A cold-worked yield strength of 250 MPa and a ductility of 12 %EL are desired. Furthermore, the final diameter must be 11.3 mm. Explain which subsequent processing steps must be foliowed to accomplish this. Hint: lt can not be done in 1 processing step!

8.3. To get weight savings, a steel is ~uenched and tempered, strengthening it trom 100 MP a, where K1c is 76 MPa·m 1

, to 2000 MPa, where K1c = 26 MPa·m 112.

What change in the critica! crack length for brittie fracture occurs for a design stress of 800 MPa?

8.4. The yield point for an iron that has an average grain diameter of 0.05 mm is 135 MPa. At a grain diameter of 0.008 mm, the yield point increases to 260 MPa. At what grain diameter will the yield point be 205 MPa?


8.5. eonsider the sugar-water phase diagram:

100

90 Solubility limit ---------.. 80

70

p 60

-- Liquid solution (syrup) ~

50 :::1 -ro ,_ (I) 0.. 40 E (I)

1-30

20

10

0 0 10 20 30 40 50 60 70

wt% sugar

a. How much sugar will dissolve in 1500 g water at 90 oe?

Liquid solution + Solid sugar

80 90

b. lf the saturated salution in part a. is caoled to 20 oe, some of the sugar will precipitate. What will be the cor'nposition of the saturated sugar salution in wt% sugar) at 20 oe?

100


8.6. Below is a portion of the H20-Nael phase diagram.

10

0 Liquid (brine)

ü 0 -~

-10 ::::::1 lee+ Salt + -~ Liquid Liquid Q)

a. E Q) I-

-20

lee+ Salt

-30 0 10 20 30

wt% salt

a. Using the diagram, explain how spreading salt on ice below 0 oe can cause the ice to melt.

b. What concentration of salt is necessary to have brine with 50 wt% ice and 50 wt% liquid at -10 oe?

8.7. A 90 wt% Sn-10 wt% Pb alloy is caoled from 300 "e to 0 oe. a. What is the composition of the alloy in terms of rnalar percentages? b. U pon cooling of this alloy, the number of degrees of treedom changes trom 2 to 1

to 0 to 1. List possîble temperatures that correspond to these tour conditîons. c. In what way does the equilibrium microstructure of this alloy ditter trom that of 90

wt% Pb-10 wt% Sn?

0

3200

3000

2800

2600

2400

2200

2000

1800

1600

1400

1200

1000


8.8. * Consider the 50 wt% Pt-50 wt% Re alloy. Th is alloy is heated to 2800 oe and then caoled to 800 oe. Perfarm complete equilibrium chemica! and physical phase analyses at 2800, 2452, 2448, and 1000 oe during the cooling process. The Pt-rich phase (Pt) is usually called a, and the Re-rich phase (Re) is usually called ~·

Weight Percent Rhenium

10 20 30 40 50 60 70 80 90 100

3186 oe

800~----~----~----~----~----~----~----~----~----~----~

0 10 20 30

Pt

40 50 60

Atomie Percent Rhenium

70 80 90 100

Re


8.9. Cast iron weighing 1 kg and containing 3.5 wt% C (remainder Fe) is melted and slowly caoled trom 1400 °C. a. Perform complete phase analyses just below 1147 oe, at 900 oe, aRd just below

727 °C. b. Sketch the expected microstructure at 500 oe.

8.1 0. The solid-state transformation from austenite to pearlite follows the Avrami equation, where y is the fraction converted and t is the time:

lt is found that y = 0.2 at 12.6 s, and y = 0.8 at 28.2 s. Determine the total time required for a conversion of 95%.

9 Changing

material

properties 11:

(Bio)composites

Composites can combine the best properties of two or more materials. A usual combination is an inorganic material {metal, ceramic) with an organic material {polymer) in order to combine strength and toughness.

Do not go upon what has been acquired by repeated hearing; nor upon traditions; nor upon rumour; nor upon what is in scripture; ... nor u pon the

consideration, "The monk is our teacher".

Siddhartha Gautama, "Buddha" in the Käläma Sutta ( c. 563-c. 483 BCE)


9.1 Structure

Composites of two materials are found in all possible combinations of metals, ceramics, and polymers.

However, one of the most important factors for the properties of a composita is the way the two materials are combined: are they just mixed at random, or is there an ordered structure? Especially in biocomposites we will find a highly optimized structure due to selection during evolution, building up by growth, and adaptation to the surroundings during their functioning.

For synthetic composites, the most common types of structures are: particles in a matrix, fibres in a matrix, and alternating /ayers. For fibres, there are ditterences between short and long fibres, and a lso the orientation of the fibres is quite important. In fact, particulate composites have much in common with materials that were obtained by precipitation hardening (see Chapter 8). Fibre-reinforced materials are aften easy to make, and are very important nowadays. Lam i nar materials are more difficult to produce.

9.2 Mechanica! properties

9.2.1 Stiffness

The modulus of a composite is dependent on the volume fraction of particles, fibres or layers of the components, and a lso on the orientation of the fibres or layers. There are two extreme cases that are easy to describe (Figure 9.1 ): - fibres or layers parallel to the applîed force: the Voigt or isostrain model; - fibres and layers perpendicularto the force: the Reuss or isostress model.

F F

F

Figure 9. 1. Left: Voigt model for composites; right: Reuss model.

The behaviour of most composites is in between these extremes, so we can predict a range for the stiffness between the outeernes of the two models.

9 Changing Properties 11: Composites 153

Voigt or isostrain model

For materials that are parallel (layers or fibres parallel to the force) the strains must be equal:

However, now the two components will feel different stresses, dependent on the surfaces they present to the force:

In total the force carried by the composite is:

Th is means for the total stress:

We now define the volume fractions in the following way:

Wethen get:

or:

When one of the components has a much higher modulus (E1 » E2), then this component now delermines the modulus of the composite (E"" V1E1). This component will also carry most of the force (F"" F1 or: cr "" V1cr1 ).

Reuss or isostress model

For materialsin series (e.g. two layers perpendicular to the force), we must have the same stress for both components:

However, each of the components has a different deformation:

The total strain is then:

öL e=-

Lo


We can introduce the volume fractions of the two materials with:

Thenwe see:

So, there is an effective modulus for the composite of:

or:

When one of the layers has a much higher stiffness (E1 » E2), then the composite modulus is determined by the component with the lowest stiffness (EI>:$ E2/V2).

9.2.2 Strength

The fracture strength of a fibre-reinforced composite is dependent on whether we have long (continuous) or short fibres.

For long fibres, the fibres must be broken, and we can estimate the strength by combining the tensile strengthof the fibres (crT,t) and the yield strengthof the matrix (crv.m) via the appropriate volume fractions:

For short fibres, fracture is determined by how well the fibres stick to the matrix (shear stress 'ttm). There is an optimum length for the fibres: too short and they break too quickly, too long and it is a waste of fibre material because they are not carrying more load. Suppose we have a cylindrical fibre with diameter dt, then at an inserted length x the area between fibre and matrix is nxd,, and the total shear force is:

F = rtdt'ttmX

The force where the fibre breaks is determined by the cross-sectional area (rcd?/4 ):

F = O'T,t rcd?/4

Matrix

9 Changing Properties 11: Composites

x

Fibre, diameter df

Figure 9.2. Forces on a fibre in a matrix.

155

F

When the total shear force equals the breaking force, we have arrived at the critica! length Xe:

When the fibres are stressed from both ends, as in the middle of a matrix, the stress is maximal in the centre of the fibre. Again, the optimum length of the fibre is reached when the maximum stress reaches the fibre tensile strength. This is twice the critica! length we just derived:

Longer fibres do not carry more load (they simply can not), so the extra fibre is wasted. Shorter fibres do not carry the maximum possible load, so the composita could be stronger.

lt is now possible to calculate the tensile strength of the composita for short fibres (the details are not shown here). Forshort fibres that are langer than the optimal length (L > 2xc), we get:

When L becomes very large, we get the formula for continuous fibers (check this!). For the optimum length fibres (l.opt = 2xc). we get trom this formula:

O'T,c = 'V:! VtcrT,f + VmO'Y,m I.C i;>". t-t V CIM U ~""J. ~. ~( k. ' 'J q .t)p ,..."J ~ 'J>r· 0 (..·

When the fibres are shorter than the optimum length (L < 2xc), the result is quite different:

However, for the optimum length, we still get the same strength (check this!). ,


9.2.3 Toughness

A composite's main advantage is that it is virtually always tougher than the materials it is made of. In a composita, a growing crack will stop at the interface of the two components. The crack will follow the interface between the materials, instead of growing into the second component, so the crack will be deflected (see Figure 9.3).

Figure 9.3. A crack in a composita can not grow straight.

Also in particulate composites, we can e.g. use rubber particles that must be torn open (absorbing a lot of energy) for a crack to grow further.

Of course, when the fibres are oriented, there will be a dependenee of the fracture strength on the direction of the applied force (see Figure 9.4 ): farces parallel to the fibres can lead to breaking of the fibres, while perpendicular farces willlead to breaking of the matrix. Finally, shear farces can lead to slipping of the fibres through the matrix.

Flgure 9.4. Fracture in oriented-fibre composites by tensi/e farces parallel (left) and perpendicular (middle) to the fibres, and by shear farces (right).

We can calcutale the work of fracture fora fibre-reinforced composita, again differentiating between long and short fibres (the details wilt nat be shown here ).

For short fibres (L < 2xc), which will be drawn out of the matrix, the interaction strength between fibre and matrix ( ttm) is important:


For long fibres (L > 2xc). the breaking of the fibres wiJl be more important:

9.2.4 Fibres

Nowadays very strong fibres can be made, such as Ultra-high molecular weight poly(ethylene) (UHMWPE or Spectra), buckytubes (C-nanotubes) and diamond fibres. C-nanotubes in theory have an ultimate tensile strength of 300 GPa, and a maximum strain of 30%. Experimentally a tensile strengthof 30 GPa has already been found, and a strain of 12%. For the Young's modulus values of ca. 1000 GPa have been observed. Take a look at Appendix C (Example 2) why this might be interesting.

The strongest fibers have a specific tensile strengthof almast 4 MPa/(kg·m-3),

and the specific Young's modulus can reach almast 250 MPa/(kg·m-3). Bath values are a bout 1 0 times higher than those for strong metals like steel or titanium alloys.

9.2.5 Compostfes as biomaterials

Composites are being used in a variety of functions, e.g. implants in joints and heart valves. Examples are: - C-fibres in a C-matrix: very strong, but hard to produce; - PMMA with hydroxyapatite and glass-ceramics as bone cement; - PE with hydroxyapatite as bone replacement; - polymers with quartz as tooth replacement; - polymers and ceramics with drugs encapsulated for controlled drug delivery; - polymers with C- or glass fibres; - Bioglass with metal (steel, Ti) fibres; - Collagraft: small (mm-sized) ceramic crystals of a mixture of hydroxyapatite and tricalcium phosphate in a collagen matrix; used for bone repair; - polyurethane with fibreglass as cast material for braken bones.

158

9.3 Blocomposites

9.3.1 Keratin

Materials Science tor Biomedical Engineering

We have al ready talked about this material in Chapter 7: it is a semi-crystalline protein that occurs in hair, nails, and the top layer of the skin (the stratum corneum or homy layer of the epidermis}. The amorphous matrix is heavily crosslinked, but does show yield. Because of this, the fracture strength (up to ca. 1 GPa) is independent of the size of notches on the surface. This is highly useful for the nails and horns that are built up mainly from keratin.

The stiffness of this composita can be calculated with the models we discussed earlier: the crystalline fibres have a modulus Et = 6.1 GPa, and the matrix has Em = 6.1 GPa in dry condition, Em = 3.1 GPa in fresh condition (20% water}, and Em = 0.9 GPa in wet condition (40% water). The volume fractions of the two components are also dependent on the amount of water: Vt = 61%, 56%, or 53% for dry, fresh, and wet, respectively (and of course: Vm = 39%, 44%, or 47%).

The Voigt model with very long fibres now gives: Ec = 6.1 GPa, 4.8 GPa, and 3.6 GPa for dry, fresh, and wet, respectively.

Experimentally it is found that Ec = 6.1 GPa, 4.3 GPa, and 1.8 GPa (dry, fresh, wet). A better agreement is found with 40-nm long fibres: = 6.1 GPa, 3.8 GPa, and 1.9 GPa tor dry, fresh, and wet.

9.3.2 Soft tissues

In general these are connective tissues such as cartilage and tendon, and the conneelive tissues around muscles and nerves. Also, part of the skin is a soft tissue. Soft tissues consist of collagen fibres in a matrix of polysaccharides and proteins (the matrix is also called "ground substance"). Usually this combination is hydrated. Often there is a fixed ratio between the collagen and the charged polysaccharides, because the interaction between charged side groups stabilizes the collagen triple helix. These tissues also contain lots of cells, and undergo growth adaptation and repair.

First we will introduce some general terms that will be used a lot. Glycosaminoglycan (GAG) has already been mentioned in Chapter 6: polysaccharides with charged groups. Proteoglycan is a combination of protein and GAG, linked by covalent bonds. When there is no covalent bond, but only hydragen bondingor electrostalie interactions, we call it a protein-polysaccharide complex.

A model that is used a lotto describe soft tissues is the lamp brush model (see Figure 9.5): collagen fibres in a matrix of proteoglycan and hyaluronic acid (a GAG). Collagen fibres are quite stift (E = 1 GPa), and can be stretched elastically up to ca. 4%. At strains of 8-10% the fibres break.

The orientation and length of the fibres is also tor these biocomposites very important: the fibres can be present in a parallel, crossed or random orientation, and they can beshort (discontinuous) or can cross the whole material (continuous).

The matrix usually behaves like a visco-elastic paste that lubricates the fibres in their movements, and can carry some stress itself.


Sulfated

mu\-polysaccharide Protein care

\ Hyaluronic acid

Figure 9.5. The "lamp brush": the matrix around the collagen fibres is a core of charged sugars, decorated with proteins carrying charged sugars.

An important property of biocomposites is the Poisson ratio (v): this can vary strongly, and is aften anisatrapie (direction-dependent). lt is aften assumed that v = 0.5 (no volume change), but for e.g. udder skin, we find v = 1, tor arterial vessel wall we see v = 0.3, and for the membrane of a pregnant locust we can even get variatien between 1 and 0, as seen in Figure 9.6.

1.0

0.5

Locust

0 0 500 1000

Strain (%)

Figure 9.6. Variation of the Poisson ratio during stretching of rubber and focust membrane.


Mesoglea

A simple example of a soft tissue is the wall of the sea anemone Metridium senile: it consistsof 6.7% collagen, 2% matrix, 86% water, and 5% salts.

The matrix is built up from neutral polysaccharides with proteins, and farms a rubbery gel with 2.4% solids. The salts screen the electrastatic interactions, which makes the gel more mobile than in the case of salt-free water.

The collagen fibres are discontinuous and oriented parallel. They give the mesoglea its strength, and lead to delayed reactions during viseaus deformation.

Mesoglea can (slowly) be deformed elastically up to 200% strain with a modulus E of ca. 1 kPa. Ouring rapid deformation, the material is more rigid (higher E). lt casts ca. 1.2 kJ/m2 to tear mesoglea, which is camparabie to rubber.

Carti/a ge

Cartilage is a very special type of connective tissue, which contains a large amount of water. lt consistsof 15-25% collagen, 1.5-10% GAG (mostly chondroitinsulphates), and 65-80% water. Sametimes there are some elastin fibres present (protein rubber). Hyaline cartilage is found on the surface of joints, between the ribs and the breast bone ( costal cartilage), and in the trachea. lt is mostly subjected to compression and shear farces. Fibrous cartilage is seen in the connections between bone and muscle, in intervertrebral discs, and in the pelvis. lt is mostly subjected to stretching, bending and twisting farces.

One of the building blocks of the matrix is PPL (2000-4000 kg/mol = 2-4 MOa), which has a protein core of 400 nm length, to which side chains of 60-100 polysaccharides are attached, each 100 nm long (50 kg/mol). Because the polysaccharides are negatively charged, they repel each other and are directed outwards. Therefore, the structure fills a very large volume (R9 = 140 nm). These PPLs aggregate via electrastatic interactions to larger complexes (50 MOa) according to the lamp brush model.

In cartilage, the collagen fibres form a network around the gel, leading to an osmotic pressure inside the material (ca. 0.35 MPa) which gives stiffness to the structure. lt has been compared to a finely woven bag with jelly inside. As an example, here are the saveral nested layers of collagen fibres with different orientations in an intervertrebral disc (see Figure 9.7).

Figure 9.7. Structure of an interverlrebral disk, with orientations of collagen shown.

Under stress, cartilage first responds elastically with a high stiffness, but after some time fluid starts leaking out, and there is stronger plastic deformation. When the

9 Changing Properties 11: Composites

stresses are removed, the opposite happens: there is a quick elastic recovery, foliowed by slow recovery due to leaking of fluid into the materia I.

161

After a long wait, one finds values like E = 16 MPa for cartilage in the ribs, and fracture occurs at a strain of 8% (stretching), or yield is found at a strain of -6% (compression). Because ofthis behaviour, cartilage can tunetion as an elastic buffer between other tissues.

Skin

Skin is a layered system: there is a hard outer layer (the epidermis of ca. 100 !lm), which contains a lot of keratin, and a dermis (2-3 mm), which is a composite of collagen fibres in a protein-polysaccharide matrix, also containing some elastin fibres. Below this is the hypodermis, which contains a lot of adipose tissue (fat cells). Th is layer is used for heat isolation, energy storage, and as elastic cushion for the bones inside. Also, the collagen fibres in this layer, although there are fewer, are conneeled to lhe conneelive tissue below the skin.

The mechanica! properties of skin are mainly determined by the dermis, so we will concentrale on that (dermis also gives strength to leather).

In the dermis, most collagen fibres are continuous through the matrix. Usually, fibres will orient in such a way that they are parallel to the local stress that the skin usually feels. There is no fixed orientation for lhe whole skin.

The matrix contains dermatan sulphate, bound to collagen fibres, and hyaluronic acid between the fibres.

Because of the elastin fibres, skin behaves like a rubber at low deformations (up to À= 1.6). At larger strains (À= 1.9) the collagen fibres become oriented in a parallel fashion, and then skin behaves more liketendon (see Figure 9.8).

12.5 1

(ll

a. 10.0 E

tJl tJl

~ 7.5 Q)

~ c: Q) 1-

5.0

2.5

1.2 1.6 2.0

Extension ratio A.

Figure 9.8. Tensile behaviour of skin.

In its normal condition, skin is pre-stressed to À= 1.1-1.3, which means that compression of skin initially only leads lo relaxation of the elastin fibres (try it!).

Skin has a very high tear energy (toughness) of 20 kJ/m2, and a stiffness of 100

MPa at high strains.


Tendon

Tendon is a composita conneelive tissue that is mainly built up from continuous, parallel collagen fibres (70-80% of the dry mass) wlth a matrix ofcetls, other proteins, polysaccharides, and salts. The matrix also contains 53% water.

Tendon can be stretched elastically toa strain of 4%, and then deforms irreversibly up to a strain of 8-10%. At the beginning of the cr,s-curve (see Figure 9.9) there is a modulus E = 1 GPa.

50

co Cl.. 40 :::2E In tJ)

~ -tJ)

30 .!!! (ii c: (I)

1-20

10

0 0 5 10

Strain (%)

Figure 9.9. De formation behaviour of tendon.

During slow stretching (2-10%/min, not very realistic) there is a curved part in the beginning of the graph, which is caused by the unfolding of the crimps in the tendon. Normally (strain rates of 20%/min) this is absent. The straight part of the graph is due to the stretching of the chains.

The tensile strength can vary between 20 and 140 MPa, but most fibres have a strengthof 50-100 MPa.

9.3.3 Hard tissues

In general, these are a composita of proteins and ceramics. In humans, the ceramic material is usually a varlation of calcium phosphate (bone). Elsewhere in nature, calcium carbonale and silica are also found.


Na ere

Nacre is not a hu man tissue, of course, but it is a very well studied example from shells. lt consists mainly of small blocks of CaC03 with an organic protein-polysaccharide matrix in between (1-4 mass%). The matrix is layered: it has an acidic protein for attachment to the ceramic, a silk-like protein, and a polysaccharide (chitin), as shown in Figure 9.10.

Aragonite platelet

Acidic protein

Fibroin-like protein

Chitin

Figure 9.10. Layered structure of nacre.

Nacre can be deformed elastically up to 1% strain with a modulus that varies between E = 60 GPa (wet) and 70 GPa (dry). The mineral has a modulus of ca. 100 GPa, and the matrix has E = 3 GPa. The models we have introduced aarlier for composites then give: E = 95 GPa (Voigt) or 39 GPa (Reuss). We see that the experimental modulus is indeed between the two extrames from the models.

Forstrains larger than 1% we get plastic deformation. Fracture only occurs at a strain of 20-30% (dry) or even 45-55% (wet). The necessary stress is 130 MPa during stretching, and 380 MPa during compression. So, the composita is very tough: the workof fracture is 400 J/m2 (dry) and 600-1200 J/m2 (wet).

This can be explained when we look at the structure: the layers of nacre are parallel to the shell surface, and contain small crystals of aragonite (0.3-0.5 llm thick). Aragonite has a modulus E = 137 GPa and a surface energy y = 0.23 J/m2

,

which leadstoa fracture toughness K1c = 0.25 MPa·m112. The tensile strengthof

aragonite is 50-100 MPa, so there is a critica! cracklengthof 2-8 Jlm. Such a crack can not fit inside the small crystals, sa fracture can only occur in the matrix.

The aragonite bleeks are shifted with respect to each other, sa a crack can nat grow straight ahead through the matrix, but must continuously switch its path, which


costs a lot of energy (see Figure 9.11 ). Th is can be compared to the way in which bricks and mortar are combined to a wall: here we also shift the bricks relativa to each other to get a strong wall.

Crack _______. opening

CaC03 "bricks"

Figure 9.11. A fracture in nacre has to grow in changing directions between the crystallites.

Bone

Boneis a very complex material: not only is it a composite, but it is also built up in a hierarchical fashion, and contains blood vessels and cells. For this latter reason, bone is a very porous material: there can be a volume fraction of 20-30% consisting of holes. This is of course bad for the strength. Further, as is typical for most biologica! materials, bone is built up during growth, and can be repaired after fracture.

As a composita, bone consists mainly of collagen as protein and hydroxyapatite (Ca10(P04)6(0H)2) as ceramic, and alsoother proteins, gtycoproteins (covalently bound proteins and sugars), and protein-polysaccharide complexes.

On the smallest length scale (0.1 J.tm), boneis a "simpte" composita of collagen fibres in a caramie matrix. Ca. 1/3 of the dry mass of boneis collagen, and its parallel fibres direct the formation of the ceramic crystals. The matrix consists of smalt (ca. 4 nm thick, 50 x 30 nm) crystals of hydroxyapatite. In young bone, there is also amorphous mineral present.

On a higher level (10 J.tm-scale) there are two types of bone: - Woven bone, with a random orientation of the collagen fibres, more mineral, and more amorphous mineral. This is the type of bone that is formed first during growth or repair. - Lamellar bone, with fibres in 5 11m thick lamellae that run in the length direction of the bone. This form is the most common one.

On even higher length scales these forms lead to two main types of bone: - Laminar bone, alternating layers of woven and lamellar bone. - Haversian bone, which is built up from cylindrical building blocks: the osteons (200 J.!ffi diameter, see Figure 9.12).

9 Changing Properlies 11: Composites

Lamellae

Figure 9.12. Structure of the osteon, with arteries, veins and nerves in the central channe/.

The ftbre orientation in the osteons is very important for the final properties of the bone, and it is therefore optimized tor the tunetion of the bone.

These types can now farm the most oomman bone structures: -Compact bone (ar cortical bone), which is best suited for bending and twisting farces.

165

- Cancellous bone (also called spongy ar trabecular bone), which occurs in places where compression farces are feit. Th is is built up from small bone columns (trabeculae), and is very porous. lt usually is covered with a layer of compact bone. The orientation of the trabeculae is optimized to carry the stress with the minimum amount of materiaL - Flat bone, a sandwich of spongy bone between two compact bone layers.

Bone is a visco-elastic material, sa there will be a dependenee on the speed of the measurements. During a rapid measurement there is an elastic deformation up to a strain of 1%, and after that there is yield. Fracture occurs at a strain of 4%. There is no necking: bone shows brittie fracture.

Boneisalso anisotropic: it is strongest in the length direction. The Poisson ratio of bone can vary between 0.13 and 0.30.

Haversian bone in humans typically has a modulus of 17 GPa during stretching, and 19 GPa in compression. The tensile strength is 220 MPa during compression.

For the various types of bone, we find elastic moduli between 5 and 30 GPa. The matrix (hydroxyapatite) has a modulus E = 130 GPa, and a tensile strength of 1 00 MP a at a strain of 0.1 %. The collagen fibres have E = 1 GPa, and a tensile strength


of 50 MPa. The models we have derived earlier for composites now predict a range for the modulus of 2.5 to 50 GPa, which is indeed correct for bone. Of course this dep&nds on the amount of ceramic matrix (ar ash content}, as seen in Figure 9.13. The amount of ceramic matrix is also important for other mechanica! properties of bone. The bending strength e.g. varies between 30 and 300 MPa, depending on the amount of caramie (50-70%).

30 Whale / 200%

20 "' [l.

<!) (il til 100% ())

~ Toughness (i.i

10

66 67 68

Mass% ash % Ash content

Figure 9.13. Left: The amount of ceramic ("ash'J determines the stiffness of bone. Right: Other properties of bone are a/so dependent on the amount of ceramic.

Tooth

Tooth is built up from various components, but for the mechanica! properties mostly enamel and dentine are important.

The enamel on the exterior of the tooth is 95% ceramic, but the inner dentine is 70% ceramic and 20% biopolymer (and 10% water). Because of this, dentine is more flexible, but also more sensitive to infections (caries) when the enamellayer is braken.

Dentine is quite similar to bone: there is a matrix of small crystals (2 x 50 x 25 nm) in a matrix of polysaccharides and collagen. However, he re there are tubules in the matrix with a strongly mineralized wall.

Tooth is mainly stressed in compression (up to 20 MPa), and that many times a day. Dentine has a Young's modulus of 12 GPa, and a compressive strengthof 200 MPa. The work offracture varies from 270 J/m2 (perpendicular to the tubules) to 550 J/m2 (parallel to the tubules).

Enamel mainly consists of ceramic material in the form of prisms. lt is much more rigid (E = 40-80 GPa) and equally strong as dentine (compressive strength 200-330 MPa). The workof fracture varies from 13 J/m2 (parallel to the prisms) to 200 J/m2

(perpendicular to the prisms).

9 Changing Properlies 11: Composites 167

lntemt.e.ZZO Enamel will dissolve at pH < 5.5, and then the dentine is attacked by bacteria and braken down. The usual salution is to drill away the dentine, and fill it with amalgam ar a polymer. The drilling may naw be over, since the weakened dentine can be selectively dissalved by a chemica! treatment at very high pH (ca. 11 ), where the enamel does not dissolve. After that, the hole can be filled as usual.

168 Matefiats Science for Biomedical Engineering

Problems

9.1. A composita consists of 40 vol% continuous E-glass fibres in an epoxy matrix. Theelastic modulus of the glass and epoxy are 76 and 2.7 GPa. a. Calculate the composita modulus parallel to the fibre axis. b. What is the composita modulus perpendicular to the fibre axis?

9.2. lt is desired that the fibres support 90% of the load in a composita containing aligned, continuous high-modulus carbon fibres embedded in an epoxy matrix. What volume fraction of fibre will be required? For epoxy assume p = 1.3 Mg/m3 and E = 3 GPa.

9.3. The Young's modulusfora composita containing 60 vol% continuous aligned fibre is 44.5 GPa when tested along the fibre axis, but only 7.4 GPa when tested normalto the fibre axis. Delermine the Young's modulus for the fibre and for the matrix.

9.4. A composita containing 25 vol% aligned continuous carbon fibres in an epoxy matrix fails at a tensile stress of 700 MPa. lf the fibre tensile strength is 2.7 GPa, what is then the matrix yield strength?

9.5. In an aligned fibreglass composite the tensile strengthof the 10-tJm diameter E-glass fibres is 2.5 GPa, the matrix has a modulus of 2.3 GPa and a yield strength of 6 MPa, and the interfacial shear strength is 0.1 GPa. a. What is the optima! cut fibre length? b. lf chopped glass fibre of optimallength is used, what volume fraction should be

added to the matrix to produce a composita tensile strength of 1 GPa? c. What is the workof fracture for the composite in b.? d. Calculate the Young's modulus for the composita in b., assuming that the force is

in the direction of the fibres. e. Calculate the fracture toughness for the composita in b., and compare this to the

toughness of the matrix (ca. 0.5 MPa·m 112).

9.6. A piece of dog skin (20 mm long, 4 mm wide and 3 mm thick) is used in a tensile test. The force increases first from 0 to 5 N, up toa strain of 20%. After that, the force rises to 130 N at 50% strain, where the skin breaks. a. Calculate the tensile strength of this skin. b. Calculate the Young's moduli for the first and second regionsof the test (assume

linear graphs ). c. Explain the presence of two reg i ons in terms of molecules. d. Calculate the toughness of this skin.


9.7.* We wish to make an enamel-like compositafora tooth prosthesis with quartz (E = 60 GPa, crT = 48 MPa) and Kevlar fibres (12 J.tm diameter). a. What volume fraction fibres is necessary to get the samemodulus as enamel (80

GPa)? Explain which model(s) you use. For the desired tensile strength of the composita, it is found that the optimal length of the fibres is 1 mm. b. Calculate the interfacial shear stress between quartz and Kevlar. Do you expect a

strong interaction between quartz (Si02) and Kevlar (a polyamide)? c. Why don't we u se quartz for the enamel replacement? lt has about the right

modulus.


Appendices

Education is an admirable thing, but it is well to remember from time to time that nothing worth knowing can be taught.

OscarWilde (1854-1900)


Appendix A English-Dutch glossary; List of symbols

lf the Dutch word is different, it is included in parentheses. lf there is a symbol for the quantity, it is included after the description in parentheses.

- adhesion (adhesie): attraction between two different compounds (Wa [J/m2])

- anisatrapie (anisotroop): different value in different directions - annealing (temperen, herstelgloeien): keepinga material at a high temperature to change its properties - atactic (atactisch): random stereochemistry in a polymer chain • austenite (austeniet): iron-carbon phase with FCC structure (y-Fe) - bainite (bainiet): microstructure of territe and cementite with coarse grains - BCC: body-centred cubic lattice in a crystal - binary phase diagram (binair fasendiagram): phase diagram with two components • bloceramie {biokeramiek): ceramic compound occurring in a living system - biologica! material (biologisch materiaal): material originating from a living system - biomaterial {biomateriaal): material inserted in a living system - biomineralization (biomineralisatie ): formation of crystals in a living system - biopolymer {biopolymeer): polymer originating trom a living system • body-centred cubic (ruimtelijk gecentreerd kubisch): kind of crystal structure - Bravals lattice {Bravais rooster): one of 14 possible crystal structures - brittie (bros): having a low toughness and high crv; absorbs little energy befare breaking • brittie fracture (brosse breuk): fracture due to rapid crack growth in ceramics and metals - bulk: average, usually the same as interior; the surface is different - bulk composition (bulksamenstelling): average composition; the surface composition is different • cancellous bone (spongieus of trabeculair bot): very porous form of bone - cartilage {kraakbeen): biologica! composite based on collagen fibres and a strongly hydraled matrix inside the fibre network -cast iron (gietijzer): Fe/C mixture with more than 2.14 wt% C • CCT -diagram: continuous-cooling-transformation diagram; shows how the phases of a material change during cooling as a tunetion of time - cellular material (cellulair materiaal): material built up as holes and walls - cellulose: plant polysaccharide • cementite (cementiet): Fe3C ceramic in Fe/C mixtures - ceramic (keramiek): material built up with covalent, network or ionic bands • close-packed structure ( dichtstgepakte structuur): crystal structure with the maximum filling of space by atoms • cohesion (cohesie): attraction between two surfaces of the samematerial (Wc [J/m2]) - coil (kluwen): irregularly folded shape adopted by long macromolecules - cold working (koud bewerken): processing technique where the material is deformed at low temperature - collagen (collageen): protein that functions as fibre in biologica! composites ·compact bone (compact of corticaal bot): bone with a low porosity - component: one of the constituents of the mixtures in a phase diagram

Appendices

- composite (composiet): material built up from two or more different simple or monolithic materials - compression (samenpersen): reducing the length in the direction of the force - conneelive tissue (bindweefsel): biologica! composite built up from collagen fibres in a visco-elastic matrix -contact angle (contacthoek): angle that a liquid drop makes with a surface; indicative for surface tensions

173

- copolymer (copolymeer): polymer built up from two or more different monomars - covalent bond (covalente binding): strong chemica! bond based on orbital overlap - covalent network (covalent netwerk): material where all atoms are conneeled by covalent bonds - crack (breuk): small imperfection in a material that can lead to complete fracture - creep (kruip): increasing strain during constant stress - critica! crack length (kritische breuklengte ): minimum length necessary for a crack befare it can grow according to Griffith's theory (c* [m]) -critica! shear stress (kritische zwichtspanning): shear stress where layers of atoms start sliding over each other in a specific slip system ('te [Pa]) - critica! stress (kritische spanning): minimum stress necessary for a crack to start growing according to Griffith's theory (cr* [Pa]) - crosslink (vernetting): strong bond (usually covalent) between two macromolecules - crystal (kristal): material with a very regular position for the atoms or molecules • crystalline (kristallijn): material built up as crystals • crystal system (kristalstelsel): specific type of crystal structure, based on its symmetry; there are 7 systems - CsCI-structure (CsCI-structuur): kind of crystal structure • cubic structure (kubische structuur): type of crystal structure with cubic unit cells • defect (fout): irregularity in a crystallattice - deformation angle (vervormingshoek, afschuifhoek): measure of deformation during shear (y [rad]) - degree of freedom (vrijheidsgraad): parameter (usually pressure, temperature, and chemica! composition) that can be changed freely within one region of a phase diagram • degree of polymerization (polymerisatiegraad): number of monomers that form a macromolecule; the "length" of the polymer chain (DP [-]) - dentine: biologica! composita in the interior of teeth • dermis (leerhuid): layer of skin that delermines its mechanica! properties - dewetting (druppelvorming): formation of dropiets of a liquid on a surface • diamond cubic structure (diamantstructuur): kind of crystal structure • dipole-dipole interaction (dipool-dipool interactie): weak bond between molecules based on the electrastatic attraction between dipales • dislocation (dislocatie): linear defect in a crystallattice - dispersion hardening (dispersieharding): increasing the strengthof a material with large particles of another material - ductile (week): low yield strength - ductile fracture (taaie breuk): fracture due to plastic deformation and necking in metals - ductility (vervormbaarheid): measure for the permanent deformation of a material after fracture; also a maasure for toughness • edge dislocation (randdislocatie): dislocation in a crystal along a line, where the defect is perpendicular to the line

174 Materials Science tor Biomedica/ Engineering

- elastic component of modulus (opslagmodulus): for visco-elastic materials, this determines theelastic part of the behaviour (E' [Pa]) - efastic deformation (elastische vervorming): reversibfe deformation of a material, usually invalving only the stretching or compression of bands - elastin (elastine): protein with typical rubber properties - etastomer (elastomeer): rubber, polymerie material with a smalt number of crosslinks - electrostatle bond (elektrostatische binding): strong bond between ions basedon eaulombic attraction - elangation factor (rekfactor): dimensionless measure for the change in length of a material (!v [-]) - enamel (glazuur): outer layer of teeth; biologica! composite based mainly on hydroxyapatite - engineering strain ("normale" rek): strain calculated with respect to the original length of the material (e [-]) - engineering stress ("normale" spanning): stress calculated with respect to the original cross section of the material (cr [Pa]) - entanglements (verknopingen): places where long macromolecules are intertwined with each other, which hinders movements - entropy (entropie): measure for the number of possibilities a system has to distribute its energy (S [J/K]) -epidermis (opperhuid): dead outer layer of skin, mostly consisting of keratin - eutectic (eutecticum): region in a phase diagram where three phases exist simultaneously - face-centred cubic (vlakgecentreerd kubisch): kind of crystal structure - FCC: face-centred cubic lattice in a crystal - territe (ferriet): Fe/C phase with a BCC structure (a-Fe) - fibre (vezel): material in a shape with a length far larger than its diameter; can be built up from oriented macromolecules -flat bone (plat bot): type of bone that is a sandwich of cancellous bone between two layers of compact bone - flexible (slap): low Young's modulus - flexural stiffness (buigingsstijfheid): modulus for elastic deformation during bending - fracture toughness (breuktaaiheid, scheurweerstand): measure for the resistance of a material against crack growth - free volume (vrij volume): difference in volume between glassy polymer and undercooled molten polymer - GAG: glycosaminoglycan -gel: salution with a very high viscosity, e.g. due to dissolved macromolecules - glass (glas): material withstrong covalent or ionic bonds, but without order; also: thermoplast or etastomer below Tg - glass (transition) temperature (glas(overgangs)temperatuur): temperature where the glass transition occurs; dependent on the speed of maasurement (Tg [KJ) - glass transition (glasovergang): process where a material changes from a glass toa liquid - glycosaminoglycan: polysaccharides with charged groups - grain (korrel): a single-crystal region in a polycrystalline material - grain boundary (korrelgrens): boundary between different grains in a polycrystalline material

Appendices 175

- hardness (hardheid): empirica! material property - Haversian bone: type of compact bone built up from osteons - HCP: hexagonal close packed lattice in a crystal - hexagonal structure (hexagonale structuur): type of crystal structure with hexagonal unit cells - hexagonal close packed structure (hexagonale dichtstgepakte structuur): kind of crystal structure - hierarchical material (hiërarchisch materiaal): material built up differently on different length scales - Hooke's law (wet van Hooke): linear relation between stress and strain - hydrogen bond (waterstofbrug): weak bond between molecules based on dipole-dipole interaction when N-H, 0-H, or F-H groups are present - hysteresis (hysterese): occurs when a process does not follow the sameroute on reversal; usually leads to the loss of energy as heat - internal friction (interne wrijving): loss of energy during deformation due to movements of macromolecules along each other -ion: electrically chargedatomor molecule - isostrain model: also known as Voigt model; describes the stiffness of materials in parallel - isostress model: also known as Reuss model; describes the stiffness of materials in series - isotactic (isotactisch): macromolecule with the same stereochemistry in each repeating unit - isotropie (isotroop): the same value in all directions - keratin (keratine): hierarchical protein material built up from a-helices - lamellar (lamellair): built up in layers - lamellar bone: farm of bone built up from small lamellae - laminar (laminair): built up in layers - laminar bone: farm of bone built up from alternating layers of woven and lamellar bone - lamp brush (flessenborstel): model for the structure of the composita soft tissues - lattice (rooster): regular distribution of lattice points inspace - lattice parameter (roosterconstante): lengthof the sides of the unit cell (a or c [m]) - lattice point (roosterpunt): mathematica! point in a lattice where one or more atoms can be located - lever rule (hefboom regel): rule that delermines how much of the two phases is formed in a two-phase region of a phase diagram - line defect (lijnvormige fout): defect that farms a line in a crystal lattice - liquidus: line in a phase diagram that indicates the composition of a liquid phase in a two-phase region -loss component of modulus (verliesmodulus): for visco-elastic materials, this delermines the viseaus part of the behaviour (E" [Pa]) - martensite (martensiet): microstructure of steel that is formed during very quick cooling - macromolecule (macromolecuul): large molecule built up from repeating monomer units; basic structure of polymers - macroscala (macroschaal): length scale in the range of practical materials (> 1 mm) -maximum strain (breukrek): strain at which fracture occurs (~>max [-])


-mechanica! properties (mechanische eigenschappen): properties of a material that determine its behaviour under stress - mefting temperature (smelt-temperatuur): temperature where a solid changes into a liquid (Tm [K]) - mesoscale (mesoschaal): length scale in the range of typical defects (1 0 nm-0.1 mm) - metal (metaal): material built up from atoms that are kept tagether by a metallic bond - metallic bond (metaalbinding): strong chemica! bond where the atoms share their valenee electrans in a common "electron sea", a molecular orbital that spans the entire material - microscala (microschaal): length scale in the range of atoms and bonds (0.1-1 nm) - microstructure (microstructuur, textuur): structure of grains in a polycrystalline material - Milier indices: numbers that indicate directions or planes in a crystal lattice - molecule (molecuul): set of atoms kept tagether by covalent bonds - monolithic material (monolitisch materiaal): material built up from a single compound; has the same chemica! composition everywhere in the bulk - morphology (morfologie, textuur): microstructure of a material - nacre (paarlemoer): biologica! composita in shells - necking (insnoering): sudden reduction in cross-sectional area of a material after yield and plastic deformation - network (netwerk): molecules or atoms that are connected with multiple covalent bonds - number-averaged molar mass (aantalgemiddelde molmassa): average molar mass of a polymer mixture, basedon mole fractions (Mn [kg/mol]) - osteon: basic cylindrical structure of Haversian bone - particulate (deeltjesversterkt): used fora composita material with particles in a matrix - pearlite (perliet): microstructure of steel with fine layers of territe and cementite - phase (fase): region in a material with a specific chemica! composition and/or aggregation state - phase angle (fasehoek): measure for the difference between the behaviour of stress and strain in the cyclic deformation of a visco-elastic material - phase diagram (fasendiagram): diagram that indicates what phases a material adopts as a tunetion of e.g. temperature and chemica! composition - phase rule (fasen regel): rule that delermines how many degrees of treedom are available for a system with a certain number of phases - planar defect (oppervlaktefout): defect in a crystal structure along a plane, e.g. a grain boundary - plastic deformation (plastische vervorming): irreversible deformation, where the bonds between atoms or molecules are braken, and reformed between other atoms or molecules - plasticizer (weekmaker): small molecule that is added to a polymer material to change its mechanica! properties; usually softens the material - point defect (puntdefect): defect in a crystal structure on one site, e.g. a vacancy - Poisson ratio (constante van Poisson): dimensionless measure for the change in cross-sectional area during stretching or compression (v [-]) - polycondensation (polycondensatie): reactions to synthesize macromolecules that involve release of water (formation of esters, amides, etc.)

Appendices

- polycrystalline (polykristallijn): material built up trom different single-crystal grains that stick tagether - polydispersity (polydispersiteit): maasure for the spread in rnalar masses in a mixture of macromolecules (D [-]) - polymer (polymeer): material built up from macromolecules

177

- polymerization (polymerisatie): chemica! reactions that couple rnanamers tofarm a macromolecule - polymorphic (polymorf): having more than one crystal structure, depending on the circumstances during crystallization - polysaccharide: polymer of sugar rnanamers - PPL: protein care with polysaccharides attached; building block of the cartilage matrix - precipitation hardening (precipitatieharding): strengthening a material by inserting small crystals of another material - protein (eiwit): polymer of amino acids - protein-polysaccharide complex (eiwit-polysacharide complex): complex between proteins and polysaccharides without covalent bonds - proteoglycan: combination of protein and GAG, linked by covalent bonds - radical polymerization (radicaalpolymerisatie): reaelions to synthesize macromolecules that involve radicals \ - radius of gyration (gyratiestraal): measure tor the size of à'r~dom coil (R9 [m]) - random-coil model (kluwenmodel): model to describe the behaviour of a macromolecule without a regular structure -recovery (herstel): after plastic deformation, there willaften be a smallelastic part of the deformation that shortens the material after the force is removed - resilience (veerkracht): energy that can be absorbed reversibly (Ee1 [J/m3

]}

- resin (hars): common name for thermoharders - Reuss model: describes the stiffness of materials in series • rock salt structure (steenzoutstructuur): kind of crystal structure • root-mean-square value (gemiddelde van de som van de kwadraten): way to average random values - rubber: elastomer, polymer with a small number of crosslinks • salt (zout): material built up trom ions and kept tagether by electrastatic bonds; all inorganic salts are caramies • SC: simple cubic lattice in a crystal • screw dislocation (schroefdislocatie): dislocation in a crystal along a line, where the defect is parallel to the line • shear modulus (glijmodulus): material proparty that indicates the stiffness during shearing ( G [Pa]) - shear stress (afschuifspanning): stress that runs paralleltoa surface (• [Pa]) - shearing (afschuiven): deforming a material with a force paralleltoa surface - silk (zijde): strong protein, used by spiders as construction material • simple cubic (simpel kubisch): kind of crystal structure ·single crystal (éénkristal): material built up from one grain; one crystallattice spans the entire material -slip system (glijsysteem): a plane and a direction in a crystal structure which define the movement of layers of atoms • solid solution (vaste oplossing): presence of isolated strange atoms in another solid


- solidus: line in a phase diagram that indicates the composition of a solid phase in a two-phase region - spheruHte (sferuliet): sphericat structure found in polymers built up from crystalline lamellae and amorphous regions - spreading coefficient (spreidingscoêfficiênt): property that delermines whether a liquid will form a film on a surface -steel (staal): Fe/C mixture with less than 2.14 wt% C • stiff (stijf): high Young's modulus - stiffness (stijfheid): measure for the force necessary to reach a certain elongation during elastic deformation of a piece of material; related to the Young's modulus of the material - storage component of modulus (opslagmodulus): for visco-elastic materials, this delermines the elastic part of the behaviour (E' [Pa]) • strain (rek): dimensionless measure for the change of the length of a material (,; [-]) • strain hardening (koudeverharding): increasing resistance against yield after stretching a material and testing it again ·stress (spanning): force per unit area (a [Pa]) -stress concentration (spanningsconcentratie): at the tip of a crack, a larger stress is found than applied from the outside ·stress intensity (spanningsintensiteit): combined measure for both the stress and crack length in a material (K [Pa·m 112

])

·stress relaxation (spanningsrelaxatie): decreasing stress at constant strain - strong (sterk): usually indicates high tensile strength, butmayalso mean high yield strength - surface composition (oppervlakte-samenstelling): chemica! composition of the surface, as opposed to the bulk composition - surface tension (oppervlaktespanning): excess energy on a surface due to the missing bonds (y [J/m2

] or [N/m]) • syndiotactic (syndiotactisch): macromolecule with alternating stereochemistry in each repeating unit - tensile strength (treksterkte): maximum engineering stress that can be carried by a material (aT [Pa]) - tension (uitrekken): increasing the lengthof a material in the direction of the force ·thermoharder: polymer that forms a covalent network after processing, and then can not be processed again ·thermoplast: polymer that is built up from macromolecules without crosslinks; can be molten and processed repeatedly -tie line (conode): constant-temperature line in a two-phase region of a phase diagram that indicates the compositions of the two related phases - tough (taai): absorbs a lot of energy before fracture; high toughness - toughness (taaiheid): area under stress-strain curve ([J/m3

]), or fracture toughness (K1c [Pa·m 112

]); also reflected in ductility • true strain (ware rek): strain calculated at each moment with respect to the immediately preceding lengthof a material (ct[-]) - true stress (ware spanning): stress as defined by the actual cross-sectional area during deformation (at [Pa]) - TTT -diagram: diagram that îndicates how the Transformation of a material is influenced by Time and Temperature -unit cell (eenheidscel): repeating unit in a crystallattice

Appendices

- van der Waals attraction (van der Waals attractie): very weak bond between molecules based on induced dipoles - visco-elastic (visco-elastisch): mechanica! behaviour is time-dependent, and determined by both elastic deformation and plastic deformation (viscous flow) - Volgt model: describes the stiffness of matenals in parallel - weak (zwak): usually indicates low tensile strength; may also mean low yield strength

179

- weight-averaged molar mass (gewichtsgemiddelde molmassa): average molar mass of a polymer mixture, based on mass fractions (Mw [kg/mol]) - wetting (bevochtigen): formation of a film of a liquid on a surface - whisker (haartje): very thin and long single crystal - work hardening (koudeharding): increasing the strength of a material by cold working - workof fracture (breukenergie): energy necessary for crack growth (Wr [J/m2

]}

-woven bone (plexiform bot): form of bone with random orientation of the collagen fibres and some amorphous mineral - yield (vloei, zwichten): beginning of plastic deformation after elastic deformation - yield strain (rekgrens): strain where yield begins (ev [-]) - yield strength (zwichtsterkte): stress where yield begins as material proparty (crv [Pa]) - yield stress (zwichtspanning): stress where yield begins (crv [Pa]) - Young's modulus: material property that indicates the stiffness during elastic deformation (E [Pa]) - zinc blende structure (zinkblende of sfaleriet-structuur): kind of crystal structure


List of symbols

Roman A [m2] A [J/mm]

A'[m2J

Ao [m] 5-l[-] a [m] a[-] B [J/m'1 $ [-] b [m] 6 [-] C[-] Cr [Pa·kg/mol] C9 [Pa·m 112

]

Cr[-] CT [K·kg/mol] c [-] c [m] c [m] c* [m] c [-] DH DP [-] d[m] d1 [m] d9 [m] E[Pa] E'[Pa] E"[Pa] Ecrack [J/m] Eel [J/m3

]

Ep [Pa] Es [Pa] e [C] F [N] F'[N] .r [-] Fa [N] f[-] ti[-] G [Pa] h [-] i[-] j[-] K [Pa·m 112

]

area constant for attraction between atoms area of tilted cross sectîon original area intercept of x-axîs in unit cell length of unit cell side parameter for line ar plane in unit cell constant for repulsion between atoms intercept of y-axis in unit cell length of unit cell side parameter for line or plane in unit een number of components in phase diagram constant in relation of aT vs. Mn constant in relation of crv vs. d9 correction factor for <r/> constant in relation of T9 vs. Mn intercept of z-axis in unit een length of unit cell side crack length (surface) or half of crack length (internal) critica! crack length in Griffith theory parameter for plane in unit cell polydispersity degree of polymerization distance between planes in a crystal fibre diameter grain diameter Young's modulus elastic or starage modulus loss modulus energy of growing crack resilience Young's modulus due to primary (covalent) bands Young's modulus due to secondary (weak) interactions charge of proton force component afferee degrees of treedom in phase diagram force between atoms fraction of covalent contribution to E fraction (mole ar weight) of phase i in total shear modulus Milier index Milier index number of repeating units in a macromolecule stress intensity

Appendices

Kp1 [Pa] constant of plastic deformation K1c [Pa·m 112

] fracture toughness k [-] Milier index ks [N/m] spring constant L [m] length Lapt [m] optimumlengthof fibre Ls [m] segment length in random-coil model Lx [m] length in x-direction Ly [m] length in y-direction Lz [m] length in z-direction L0 [m] originallength Mj [kg/mol] molar mass of macromolecule with j monomars Mn [kg/mol] number-averaged molar mass Mw [kg/mol] weight-averaged rnalar mass Ma [kg/mol] rnalar mass of repeating unit m [-] exponent of attraction between atoms N [mol] total number of macromolecules NA [mol-1

] Avogadro's number: number of particles in a male Nj [mol] number of macromolecules with OP = j N [-] strain hardening exponent in plastic deformation n [-] exponent of repulsion between atoms ns [-] number of segments in random-coil model P [-] number of phases R [m] interatomie distance R0 [m] equilibrium interatomie distance R1 [m] parameter in Orowan model R9 [m] radius of gyration r [m] radius of atom re [m] distance between chain ends r+ [m] radius of cation r_ [m] radius of anion <rc2> [m2

] root mean square of re S [ J/K] entropy Si; [J/m2

] spreading coefficient of jon i T [KJ temperature T9 [KJ glass transition temperature Tg,oo [K] Tg at infinite mol ar mass Tm [K] melting temperature t [s] time U [J] energy of pair of atoms or ions Uo [ Jj U at R = Ro V[m] volume V; [-] volume fraction of substance i V0 [m

3] original volume

Wa,ii [J/m2] ad hesion between substance i and j

Wc,i [J/m2] cohesion of substance i

Wt [J/m2] work of fracture

w [J/m3] absorbed energy per cycle wi [-] mass fraction of macromolecules with OP = j x[-] x-coordinate

181

182

Xe [m] Xi (-]

Xj[-] Xo [-]

y [-] z [-] z_ [-] Z+ [-]

Greek u [rad] as [rad] ~[rad] ~s [rad] 1 [rad] 1 [rad] 1 [J/m2] 1i [J/m2] 1ii [J/m2] o [rad] s[-] Be[-] ~>max [-] €t [-]

€x [-]

€y [-]

€y [-]

l>z [-]

€0 [F/m] ll [Pa·s] e [rad] À[-] V(-) v [Hz] p [kgtm1 Pc [m] cr [Pa] cra [Pa] crc [Pa] crint [Pa] Omax [Pa] crr [-] crT (Pa] crT,oo [Pa] crt [Pa] crv [Pa] crv.sc [Pa] cr* [Pa]

Matenals Science tor Biomedical Engineering

critica! fibre length composition (mole or weight fraction) of phase i mole fraction of macromolecules with OP = j average or initial composition of whole system (mole or weight fraction) y-coordinate z-coordinate charge of anion as a multiple of e charge of cation as a multiple of e

angle in unit cell angle in slip system angle in unit cell angle in slip system angle in unit cell deformation angle during shear surface energy surface energy of substance i versus air surface energy of interface between substance i and j phase angle strain constant strain during stress relaxation maximum strain at breaking, or during dynamica! testing true strain strain in x-direction yield strain strain in y-direction strain in z-direction dielectric constant of vacuum viscosity arbitrary angle elongation Poisson ratio frequency density radius of curvature at crack tip tensile stress externally applied stress constant stress during creep intemal stress at crack tip maximum stress in Orowan model, or during dynamica! testing characteristic radius of polymer coil tensile strength tensile strength at infinite molar mass true stress yield stress yield stress of single crystal critica! stress in Griffith model

Appendices

1: [Pa] shear stress 'te [Pa] 'tfm [Pa] 'tK [s] 'tmax [Pa] "CR [s]

critica! shear stress in single crystal interfacial shear strength for fibre and matrix

'tr [s] Ijl [rad] ro [rad/s]

creep relaxation time for linear elastic solid ('tcr in the 181 edition) theoretica! maximum shear stress stress relaxation time for linear elastic solid ('te in the 181 edition) relaxation time in Maxwell and Voigt models angle between veetors angular frequency

Only In Appendices A9 [m

2} area of ground surface

bs [m- ] constant in random-coil model Cs [.1/K] constant in entropy equation ~ [Pa] slope of cr,~::-curve Etract [J] energy for fracture in Orowan model Fe [N] centripetal force F9 [N] gravitational force G [Nm2kg2

] gravitational constant g [m/s2

] gravitational acceleration at the Earth's surface ke [J/K] Boltzmann constant Mffl [kg] mass of Earth Ne [m-] chain density h [m] height above surface hmax [m] maximum height of tower hs [m] height of geostationary orbit L [m] length of cable m [kg] mass of satellite Ne [-] coordination number in ionic structures Q [J] heat r [m] radius of orbit rs [m] radius of geostationary orbit r® [m] radius of Earth t [0 C] temperature in degrees Celsius t' [°F] temperature in degrees Fahrenheit U [J] internal energy W[J] work W(rc) [m-1

] distribution function related to chain end distance W(x,y,z) [m-a] distribution function related to position ASac [J/K] entropy of all chains ASse [J/K·m1 entropy of single chain AStot [J/K] total entropy of rubber crc [Pa] compression strength of brick

183


Only in lntermezzos A[-] CL [J/K] Ge [J/m2

]

K[Pa] i[-] n [-] n [-] t[-] u[-] V[-) w[-] Yc [J/m2

]

YP [J/m2] ev [-] rJo [Pa·s] À [m]

Madelung constant heat capacity during elangation critica I strain energy release rate, or toughness bulk modulus Miller-Bravais index for planes in a hexagonal unit cell sealing number for Miller-Bravais indices integer number in X-ray diffraction Miller-Bravais index for directions in a hexagonal unit cell Miller-Bravais index for directions in a hexagonal unit cell Miller-Bravais index for directions in a hexagonal unit cell Miller-Bravais index for directions in a hexagonal unit cell surface energy energy necessary for plastic deformation during fracture strain based on volume change standard viscosity wavelength of X-rays

Appendices

Appendix B More on true stress and strain

The force on a material duringa tensile test is: F = crtA

We will work with the true strain (et) here.

The maximum force is reached when: d F = 0 = A d cr 1 + cr 1 d A ds1 ds 1 ds1

Now with constant volume (V= AL):

dA Therefore:

dE1

Since s1 1{ ~) we get: dL 1 dL , and: -=L

L de1

So: dA -A, and therefore: A= A0 e-"• dt1

With crt = Kpl&tN, it then follows that: dF = A(KP1 Nt~ 1 Kp1~>~)= KP1Ae~-1 (N-e 1 ) det

The maximum force is found when d F 0 , which means: dEt

This is where necking occurs.

185

Before necking e1 < N, so dF > 0: true stress increases faster than A decreases, so det

the material can carry a larger force F, or engineering stress cr = F!Ao.

After necking Et> N, so dF < 0: A decreasas taster than true stress increases, so d~:: 1

the material can carry a smaller force or engineering stress.

We will now look at an ideal elastic material, which has a linear relation between true stress and true strain: crt = Eet

lt then follows from the relations above: cr _ln(1+e) E 1+E


When we plot ~ vs. E, this automatically gives a maximum in cr during tension, E

because the cross-sectionat area decreases faster then the force increases. This does not happen during compression.

alE 0.4

0.2

-2

-0.2

-0.4

-0.6

2 4 6 8

The maximum remains even when you plot cr versus ~;1 . You will then get: ~ = ~; 1 e-"' E

alE 0.4

0.2

-1

-0.2

-0.4

-0.6

1 2 3 4

Appendices 187

The modulus is defined as: E = ~. but in terms of cr and e this becomes: Et

Ê dcr = E 1 ln(1+e) de (1+e)2

Platting ~ vs. e then gives:

ÊIE

The modulus seems to increase during compression, and seems to decrease (and even become negative) during tension. These are all artetacts from the definitions of cr and e.


Appendix C Examples of towers

In order to show some practical applications of material properties, we willlook at two constructions that go to the limit.

Example 1: Brick tower Brick has a density of 2000 kg/m3

, and therefore applies a pressure downwards per meter of height of 19600 Pa (= pg) on the underlying brick. The compression strength (ac) of brick is 40 MPa, so the maximum height of a brick tower with constant crosssectional area is:

hmax = ~ = 40x1 06/19600 2 km pg

When the tower is higher, the pressure on the lowest layer of bricks (which carries the rest) is higher than the strength of the materia I, sa the bottorn layer will crumble. This is already a quite nice Tower of Babel.

The calculation becomes more complex when we want to build a pyramid instead of a straight tower: the lowest layer of bricks then carries a much lower weight. The maximum allowed pressure on the lowest layer is then:

Here, V is the volume of the tower, and Ag is the ground surface area. We can then write:

V = ac = 2 km Ag pg

When we build a pyramid of height h, we get: V= A9h/3, so:

hmax = 6 km

This is quite close to the actual height of Mount Everest (rock instead of brick), and also explains why it is no surprise that the pyramid has been invented a number of times in different places and times.

Appendices 189

Example 2: Space elevator Fora satellite of mass mmoving in a stabie orbit of radius rwith an angular speed co around the Earlh, there must be a centripetal force {Fe= mco2r) which is due to the gravitational force between the satellite and the Earlh (F9 = GMfflm/r2

).

A geostationary satellite always hangs over the same point of the equator, so it must have a revolution time of 24 hours. With G = 6.67·10-11 Nm2kg2

, mass of the Earlh Mffl = 6·1024 kg and co= 21t/(24x3600) = 7.3·10-5 rad/s, the geostationary orbit has a radius:

r GM J113 rs = l co 2® = 4.23·107 m = 42300 km

Since the Earlhitself has a radius r6 = 6.4·106 m, the height above the Earlh's surface then is:

hs = rs- (<JJ = 36000 km

From such a satellite, one can lower a cable to the Earlh's surface, as long as a cable is also deployed on the other side of the satellite (away trom the Earlh), in order to keep the centre of mass in the geostationary orbit. Then, one can use a simple elevator system along the cable to go to space: the space elevator. Rockets will no longer be necessary.

The only problem is: can a cable that is 36000 km long or more carry its own weight? This problem is more complex than the tower in the previous case, since the value of the acceleration g is not constant anymore.

Let's assume that we use a cable with constant cross-sectional area A. On each small piece of the cable with length dr, at a radius rfrom the centre of the Earlh, there are stresses in the cable towards the Earlh, a{r), and away from the Earlh, a(r + dr). Now, the centripetal force must be generaled by the gravitational force and these stresses:

Fe= co2rApdr = F9 + a(r)A- a(r + dr)A

Th. . a(r+dr)-a(r) da GMI!:!p 2 IS QIVeS: = = -- - CO pr

dr dr r 2

The maximum stress in the cable is reached when da = 0, so at r = r5 • This means dr

that the entire weight of the cable hangs from its point in the geostationary orbit. When we use the boundary condition that the cable end at the Earlh's surface must hang free without stress (a(r<JJ) = 0), we get:

( 1 1 r~- r

2 J a(r) = GMep - + 3 r0 r 2r

5


When we now plot cr vs. r , we get the following curve: GMEBp

5 10 15 20 25

The totallength of the cable (L) now follows from the boundary condition that the other end of the cable must a lso hang tree in space without stress:

cr(rEB + L) = 0

Aftersome number crunching, this gives: L = 144000 km, or 101700 km outside of the geostationary orbit and 42300 km within the orbit down to the surface. The total mass required for this cable is ALp.

lf we go back to the maximum stress that the material has to carry, we see that:

C>max = cr(rs) = 48.4p MPa

For realistic materials, the value of is on the order of 1-2 MPa/kg·m-3, so we

p can not directly start to build the space elevator.

An obvious salution is of course to make the cross-sectional area of the cable large in the geostationary orbit (where the highest laad is carried), and smaller elsewhere in order to decrease the mass to carry. We can e.g. choose to keep the stress constant everywhere in the cable. Then it turns out that we have to use a cross-sectional area with an exponential decrease when we move away from the

geostationary orbit. When we use carbon fibres as material (very high crmax ), we can p

build a realistic space elevator with a cross-sectio na I area that is 10 times smaller on the Earth's surface than in geostationary orbit.

Appendix D Literature

Appendices

- J.E. Gordon, The new science of strong materials (HAB76GOR): Very good explanation of the principles of materials science.

191

- J.E. Gordon, Structures (FEB91 GOR): Application of materials science in constructions; does not really belang to the topics in this book, but is very readable. - J. Vincent, Structural biomaterials (PCW90VIN): Biologica! materials seen in a mechanica! way; very good overview. - S.A. Wainwright et al., Mechanica! design in organisms (PCW76MEC): Good overview of the mechanica I properties of various biologica! materials; goes on with structures in organisms. - J.B. Park & R.S. Lakes, Biomaterials (PCW92PAR): Properties ofvarious biomaterials. -LV. lnterrante & M.J. Hampden-Smith, Chemistry of advanced materials, Ch. 11: e.c. Perry, Biomaterials (HDS98CHE): Recent developments concerning biomaterials. -Van der V egt, Polymeren: van keten tot kunststof (HSB96VEG); in Dutch, good overview of polymers. - R. Young, Introduetion to Polymers (HSB91YOU): More thorough than van der Vegt on polymers. - P. Bali, Made to measure (HAB97BAL): Readable account of recent developments in materials science. - D. Bloor et al., Encyclopedia of advanced materials (4 parts, HAA94ENC): An encyclopaedia, need I say more? - Y. Fung, Biomechanics-Mechanical properties of living tissues (PFE93FUN): Contains a more elaborate section on visco-elasticity, bone and muscle.


Appendix E Selected material properties

Table 1. Modulus of elasHclty (E), shear modulus (G), Poisson ratio (v), density (p), and reduced modulus (E/p) of various materials at room temperafure

E G V p Elp [GPa] [GPa] [-] [Mg/m3

] [MNm/kg] eerom/es Na Cl 49 2.17 22.6 LiF 112 MgO 293 Si02 94 37.6 0.25 2.6 36.2 Pyrexglass 69 28.8 0.20 2.23 30.9 AI203 390 154.8 0.26 3.9 100 Zr02 200 5.8 34.5 Fireclay brick 97 1.5-1.8 Covalently bound eerom/es SbN4 310 3.2 SiC 450 2.9 155 TiC 379 7.2 52.6 wc 550 225.4 0.22 15.5 35.5 Diamond 1050 0.20 3.51 285 Si 166 Ge 129 Metals Al 70 26.3 0.33 2.7 25.9 Al-alloys 72.4 27.6 0.31 2.7 26.8 Cu 115 8.94 12.9 Cu-alloys 117 44 0.33 8.9 13.1 Ni 207 77.7 0.30 8.9 23.2 Fe 205 7.87 26.0 Steels 207 77.7 0.33 7.8 26.5 Stainless steel 193 65.6 0.28 7.9 24.4 Ti 110 44.8 0.31 4.5 24.4 w 384 157 0.27 19.3 20 Polymers PE 0.4-1.3 0.4 0.91-0.97 ca. 0.96 PMMA 2.4-3.4 1.2 ca. 2.4 PS 2.7-4.2 0.4 1.1 ca. 3.2 Nylon 1.2 0.4 1.2 ca. 1 Rubbers 0.01-0.1 0.49 1.2-1.6 Varlous Concrete-cement 45-50 0.20 2.5 ca. 19 Common bricks 10-17 Gomman wood

(11 grain) 9-16 0.4-0.8 ca.20 (.lgrain) 0.6-1.0 0.4-0.8 ca.1.3

Granite 40 2.4

Appendices 193

Table 2. Typical values for plastic deformation (crt = KplstN) of metals at room temperafure

Kp1 [MPa] N [-] Al Alloys

1100, annealed 180 0.20 2024, hardened 690 0.16 6061, annealed 205 0.20 6061, hardened 410 0.05 7075, annealed 400 0.17

Brass (Cu-Zn) 70-30, annealed 895 0.49 85-15, cold-rolled 580 0.34

Bronze (Cu-Sn), annealed 720 0.46 Co-alloy 2070 0.50 Cu, annealed 315 0.54 Mo, annealed 725 0.13 Steel

low-carbon, annealed 530 0.26 1045, hot-rolled 965 0.14 4135, annealed 1015 0.17 4135, cold-rolled 1100 0.14 4340, annealed 640 0.15 52100, annealed 1450 0.07 302, ·stainless, annealed 1300 0.30 304 stainless, annealed 1275 0.45 410 stainless, annealed 960 0.10

Table 3. Slip systems and critlcal shearlng stress (1:c) in crystals

cc [MPa] Slip plane Slip direction c;d l

FCC (12 slip systems) {111} <110> Ag (99.99%) 0.58 Cu (99.999%) 0.65 Ni (99.8%) 5.7

Diamond cubic (12 slip systems) {111} <110> Si, Ge

sec (12 slip systems) {11 0} <111> Fe (99.96%) 27.5 Mo 49.0

HCP (3 slip systems) {0001} <112 0> Zn (99.999%) 0.18 Cd (99.996%) 0.58 Mg (99.996%) 0.77 AI203, BeO

Rock salt (6 slip systems) (110) [11 0] LiF, MgO


Table 4. Mechanica! properties of polymers

E[GPa] p [Mg/m3] crv [MPa] Tg [OC]

Thermoplastics High-density PE 0.56 0.96 30 Low-density PE 0.18 0.91 11 -20 PVC 2.0 1.2 25 80 pp 1.3 0.9 35 0 PS 2.7 1.1 50 100 Polycarbonate (PC) 2.5 1.2-1.3 68 150 PET 8 0.94 135 67 PMMA 2.8 1.2 70 100 Polyesters 1.3-4.5 1.1-1.4 65 67 Polyamide 2.8 1.1-1.2 70 60 Teflon 0.4 2.3 25 120

Thermosets Epoxies 2.1-5.5 1.2-1.4 60 107-200 Phenolics 18 1.5 80 200-300 Polyesters 14 1.1-1.5 70 200 Polyimides 21 1.3 190 350 Silicones 8 1.25 40 300 Ureas 7 1.3 60 80 Urethanes 7 1.2-1.4 70 100

Appendices

Table 5. Toughness of some materials

Metals Al Al 2024, hardened Al 2024, annealed Al 7075, hardened Ti-6A1-4V Mild steel Steel 4340, hardened Steel 4340, annealed Maraging steel Pressure vessel steel

Ceramics Soda glass Ab03 Si02 SiC SbN4 wc TiC Zr02, stabilized Concrete

Polymers High density PE PMMA PS Polycarbonate (PC) Epoxy Fibreglass reinforeed epoxy

K1c [MPa·m 112]

45 26 44 26 55

140 60.4 99

ca. 133 200

0.7 3.7 1 3.1 4

80 0.5

12 0.2

ca.2 ca. 1.3 ca. 1

2 ca. 0.5

ca.40

195


Table 6. Mechanica! properties of metals

crr [MPa] crv [MPa] Ductility Ploin oorbon steel (%elong.) 1020 (0.20C, 0.45Mn) as rolled 448 331 36

annealed 393 297 36 1040 (0.40C, 0.75Mn) as rolled 621 414 25

annealed 517 352 30 hardened 800 593 20

1080 (0.80C, 0.75Mn) as rolled 967 586 12 annealed 614 373 25 hardened 1304 980 1

Low alloy steel 4340 (0.40C, 0.90Mn, annealed 745 469 22

0.80Cr, 1.83Ni) hardened 1725 1587 10 5160 (0.60C, 0.80Cr, annealed 725 276 17

0.90Mn) hardened 2000 1773 9 8650 (0.50C, 0.55Ni, annealed 710 386 22

0.50Cr) hardened 1725 1552 10 Stalnless steel 430 (17Cr, 0.12C max, ferritic) 550 375 25 304 (19Cr, 9Ni, austenitic) 579 290 55 Wrought aluminium alloys 1100 (99.0 min Al, annealed 89 24 25

0.12Cu) half-hard 124 97 4 2024 (4.4Cu, 1.5Mg, annealed 220 (max) 97 (max) 12

0.6Mn) heat-treated 500 390 5 6061 (1.0Mg, 0.6Si, annealed 152 (max) 82 (max) 16

0.27Cu, 0.2Cr) heat-treated 315 280 12 7075 (5.6Zn, 2.5Mg, annealed 276 (max) 145 (max) 10

1.6Cr, 0.23Cr) heat-treated 570 500 8 Cast aluminium alloys 356 (?Si, 0.3Mg) sand-cast 207 (min) 138(min) 3

heat-treated 229 (min) 152(min) 3 413 (12Si, 2Fe) die cast 297 145 2.5 Wrought copper alloys OFHC (99.99Cu) annealed 220 69 45

cold worked 345 310 6 Electrolytic tough annealed 220 69 45

pitch (0.040) cold worked 345 310 6 Brass (70Cu, 30Zn) annealed 325 105 62

cold worked 525 435 8 Beryllium capper salution 410 190 60

(1.7Be, 0.20Co) treated hardened 1240 1070 4

Casting copper alloys Leaded bronze as cast 255 117 30

(5Sn, 5Pb, 5Zn) Aluminium bronze as cast 586 242 18

(4Fe, 11AI)

Appendices 197

Table 7. Properties of relnforcing fibres, matrices, and composites

p E O"T K1c [Mg/m3] [GPa] [GPa] [MPa·m112]

Fibre E-glass (10 !lm) 2.56 76 1.4-2.5 High-modulus carbon 1.75 390 2.2

(8 flm) High strength carbon 1.95 250 2.7

(8 Jlffi) Kevlar (12 !lm) 1.45 125 3.2 SiCwhisker 3.2 480 3 SiC fibre (140 !lm) 3.0 420 3.9 AI203 (3 !lm) 3.2 100 1.0 SbN4 3.2 380 2 Boron ( 1 00-200 Jlffi) 2.6 380 3.8

Matrix Epoxy 1.2-1.4 2.1-5.5 0.063 Polyester 1.1-1.4 1.3-4.5 0.060

Composites CFRP 1.5 189 1.05 32-45

(85% C-fibre in epoxy) GFRP 2 48 1.24 42-60

(50% glassin polyester) KFRP 1.4 76 1.24

(60% Kevlar fibre in epoxy)

2014-AI (20% AI203, 103 0.48 particulate)

2124-AI (30% SiC, 130 0.69 particulate)

6061-AI 69 0.31 6061-AI (51% B, 231 1.42

continuous fibre) 6061-AI (20% SiC, 115 0.48

discontinuous fibre) Si3N4 (1 0% SiC whisker) 0.45 AI203 (10% SiC whisker) 0.45

198

Appendix F Non-SI units

Length:

Materfals Science for Biomedical Engineering

1 inch (1 in. or 1") = 2.54 cm 1 foot (1 ft or 1') = 0.305 m 1 Angstnam (1 A) = 0.1 nm

Force: 1 pound-force (1 lbf) = 4.448 N 1 kilogram-force (1 kgf)= 9.808 N 1 dyne= 10 r.tN

Ma ss: 1 pound-mass (1 lbm) = 0.454 kg

Pre ss ure /stress: 1 psi (pound-force per square inch) = 6.90 kPa 1 ksi = 1000 psi

Energy: 1 cal (calorie)= 4.184 J 1 erg = 0.1 r.tJ 1 Btu (British thermal unit)= 1054 J

Power: 1 hp (horse power) = 7 46 W

Temperaf ure: TIK = tfOC + 273.15 f' / 0 f = (9/5) X (f/0 C) + 32

Molarmass: 1 Da ( dalton) = 1 g/rnol = 1 o-3 kg/mol

Appendices

Appendix G Periadie table; Physical constants; Properties of elements

H 1 u Be 3 4

Mg 12

Db Sg 105 106

La Ce Pr 57 58 59 Ac Th Pa 89 90 91

Physical constants

($ = Only in Appendices)

eo = 8.85419x1 o-12 F/m

e = 1.6021765x10-19 c

NA= 6.02214x1023 mor1

G = 6.67x10-11 Nm2kg2 ($)

g = 9.81 m/s2 ($)

kB = 1.38066x1 o-23 J/K ($)

M$ = 6x1024 kg($)

Nd 60 u 92

Mn Fe Co 25 26 27 Tc Ru Rh 43 44 45 Re Os Ir 75 76 77 Bh Hs Mt 107 108 109

Pm Sm Eu 61 62 63 Np Pu Am 93 94 95

B c N 0 5 6 7 8 Al Si p s 13 14 15 16

Ni Ga Ge As Se 28 31 32 33 34 Pd In Sn Sb Te 46 49 50 51 52 Pt Au Tl Pb Bi Po 78 79 81 82 83 84 Os Rg 110 111 112 114 116

Gd Tb Dy Ho Er Tm Yb 64 65 66 67 68 69 70 Cm Bk Cf Es Fm Md No 96 97 98 99 100 101 102

199

He 2

Ne 10 Ar 18

Br Kr 35 36 I Xe

53 54 At Rn 85 86


Properties of elements at 25 oe and 1 bar {unless stated otherwise)

lonic radius can be dependent on coordination number: . -. zinc blende= 4, rocksalt = 6, CsCI = 8. (See Appendix L for more information.)

Atomie radius: half the bond between an element and itself. a/p = atoms per point for cubic or hexagonal structures.

Element Molar Ion ie Atomie Crystal Lattice Solidor ma ss radius/nm radius Structure parameter liquid /g·mor1 (charge, /nm cubio-hex density

coordination) a, c/nm /Mg·m-3

Actinium 227.07 0.112 (3+,6) 0.188 FCC 0.5311 10.06 (ssÄC) Aluminium 26.98 0.039 (3+,4) 0.143 1 r-vv 0.4050 2.70

I (1~1) 0.054 (3+,6) Americum 243.06 0.098 (3+,4) 0.173 0.347, 13.67

~ 0.109 (3+,6) 1.124 112.76 0.076 (3+,6) 0.141 rhomb 6.69

(51Sb) ·~ 0.060 (5+,6) ArQon C1aAr) 39.95 - - gas - -Arsenic 74.92 0.058 (3+,6) 0.125 rhomb - 5.72 (33As) 0.034 (5+,4}

0.046 (5+,6) Astatine 209.99 ? 0.140 ? (ssAt} Barium 137.33 0.135 (2+,6} 0.217 BCC 0.5019 3.5 (5sBa) 0.142 (2+,8) Berkelium 247.07 0.096 (3+,6) 0.172 HCP 0.342, 14.79 (97Bk) 1.107 Beryllium 9.012 0.027 (2+,4) 0.114 HCP 0.229, 1.85 c4Be> 0.045 (2+,6) 0.358 Bismuth 208.98 0.103 (3+,6) 0.170 rhomb - 9.75 (a3Bi} 0.117 (3+,8)

0.076 (5+,6) Bohrium 262.12 ? ? ? ? <><37 C101Bh) Boron 10.81 0.023 (3+) 0.097 rhomb - 2.34 (3B) Bramine 79.90 0.196 (1-,6) 0.119 liquid - 3.14 (35Br) 0.025 (7+,4)

0.039 (7+,6) Cadmium 112.41 0.078 (2+,4) 0.148 HCP 0.298, 8.65 (48Cd) 0.095 (2+,6) 0.562

0.110 (2+,8) Calcium 40.08 0.100 (2+,6) 0.197 FCC 0.5582 1.55 C2oCa) 0.112 (2+,8) Californium 251.08 0.095 (3+,6} 0.199 HCP 0.339, ? (gaCf) 1.102 Carbon 12.011 0.015 (4+,4) 0.077 ... 0.246, 2.25 (sC) 0.016 (4+,6) \-\,f-)( 0.671 Carbon idem idem idem FCC 0.3567 3.51 (> 60 GPa)

a/p

1

1

4

--

?

1

4

2

-

?

--

2

1

4

4

2

Appendices 201

Element Mol ar Ion ie Atomie Crystal Lattice Solidor a/p ma ss radius/nm radius Structure parameter liquid /g·mol-1 {charge, /nm cubic-hex density


Cerium 140.12 0.101 (3+ ,6) 0.183 HCP 0.368, 6.75 4 (5sCe) 0.114(3+,8) 1.186

Caesium 132.91 0.167 (1+,6) 0.263 BCC 0.6080 1.87 1 (55Cs) 0.174 (1+,8) Chlorine 35.45 0.181 (1-,6) 0.107 gas - - -(17CI) 0.008 (7+,4) Chromium 52.00 0.062 (3+,6) 0.125 BCC 0.2885 7.19 1 (24Cr) 0.026 (6+,4)

0.044 (6+,6) Co balt 58.93 0.056 (2+,4) 0.125 HCP 0.251, 8.85 2 (21Co) 0.065 (2+,6) 0.407

0.090 (2+,8) 0.055 (3+,6)

Copper 63.55 0.060 (1+,4) 0.128 FCC 0.3615 8.94 1 (29Cu) 0.077 (1 +,6)

0.073 (2+,6) Curium 247.07 0.097 (3+,6) 0.174 HCP 0.350, 13.30 4 (95Cm) 1.133 Darmstadtium ca.281 ? ? ? ? ? ? (11oDs) Dubnium 262.11 ? ? ? ? 29.00 ? (105Db) Dysprosium 162.50 0.091 (3+,6) 0.175 HCP 0.359, 8.55 2 (55Dy) 0.103 (3+,8) 0.565 Erbium 167.26 0.089 (3+,6) 0.173 HCP 0.356, 9.07 2 (6sEr) 0.100 (3+,8) 0.559 Europium 151.97 0.095 (3+,6) 0.204 BCC 0.4583 5.24 1 (53Eu·) 0.107 (3+,8) Fermium 257.10 ? ? ? ? ? ? (1ooFm) Fluorine 19.00 0.133 (1-,6) 0.0709 gas - - -(gF) Francium (87Fr) 223.02 0.180 (1+,6) 0.270 ? ? ? ? Gadolinium 157.25 0.094 (3+,6) 0.179 HCP 0.364, 7.90 2 (54Gd) 0.105 (3+,8) 0.578 Gallium 69.72 0.047 (3+,4) 0.135 ortho - 5.91 -(31Ga) 0.062 (3+,6) Germanium 72.59 0.073 (2+,6) 0.122 FCC 0.5658 5.32 2 bGe) 0.039 (4+,4)

0.053 (4+,6) Gold 196.97 0.137 (1+,6) 0.144 FCC 0.4079 19.3 1 (79Au) 0.085 (3+,6) Hafnium 178.49 0.058 (4+,4) 0.157 HCP 0.319, 13.31 2 (12Hf) 0.071 (4+,6) 0.505

0.083 (4+,8) Hassium ,.,265 ? ? ? ? ,., 41 ? (10sHs) Helium 4.003 - - gas - - -(2He)

202 Materlafs Science for Biomedicaf Engineering

Element Mol ar Ion ie Atomie Crystal Lattice Solidor a/p [ mess radius/nm radius Structure parameter Hquid /g·mor1 (charge, /nm cubic-hex density


Holmium 164.93 ? 0.147 HCP 0.358, 8.80 2 (s?Ho) 0.562 Hydragen 1.008 0.0373 gas - - -(1H) Indium 114.82 0.062 (3+,4) 0.166 tetra 7.31 -(4eln) 0.080 (3+,6) lodine 126.91 0.220 (1-,6) 0.136 ortho - 4.94 -(s31) 0.095 (5+,6)

0.042 (7+,4) 0.053 {7+,6)

Iridium 192.22 0.068 (3+,6) 0.135 FCC 0.3839 22.56 1 (nlr) Iron 55.85 0.063 (2+,4} 0.124 BCC 0.2866 7.87 1 (zefe) 0.061 (2+,6)

0.092 (2+,8) 0.049 (3+,4) 0.055 (3+,6) 0.078 (3+,8)

Iron idem idem idem FCC 0.3647 ? 1 (> 912 oe} Krypton 83.80 - - gas - - -(36Kr)

• Lanthanum 138.91 0.103 (3+,6) 0.188 HCP 0.377, 6.15 4 • (s?La) 0.116 (3+,8) 1.217

Lawrencium 262.11 ? ? ? ? ? ? C10sLr) Lead 207.2 0.119 (2+,6) 0.175 FCC 0.4950 11.3 1 (szPb) 0.129 (2+,8)

0.065 (4+,4) 0.078 (4+,6) 0.094 (4+,8}

Lithium 6.94 0.059 (1 +,4) 0.152 BCC 0.3509 0.53 1 (sLi) 0.076 (1+,6)

0.092 (1+,8) Lutetium 174.97 0.086 (3+,6} 0.172 HCP 0.351, 9.84 2 Cri Lu) 0.097 (3+,8) 0.555 Magnesium 24.31 0.057 (2+,4) 0.160 HCP 0.321, 1.74 2 (12Mg) 0.072 (2+,6) 0.521

0.089 (2+,8) Mangenese 54.94 0.066 (2+,4) 0.112 BCC 0.8914 7.43 29 (2sMn} 0.083 (2+,6)

0.096 (2+,8) 0.058 (3+,6) 0.039 (4+,4) 0.053 (4+,6) 0.033 (5+,4) 0.026 (6+,4) 0.025 (7+,4)

Appendices 203

Element Mol ar Ion ie Atomie Crystal Lattice Solidor a/p ma ss radius/nm radius Structure parameter liquid /g·mor1 (charge, /nm cubic-hex density


Meitnerium 276.15 ? ? ? ? ? ? (1o9Mt) Mercury 200.59 0.119 (1+,6) 0.150 liquid - 14.2 -(soHg) 0.096 (2+,4)

0.102 (2+,6) 0.114 (2+,8)

Molybdenum 95.94 0.069 (3+,6) 0.136 BCC 0.3147 10.2 1 (42Mo) 0.065 (4+,6)

0.046 (5+,4) 0.061 (5+,6) 0.041 (6+,4) 0.059 (6+,6)

Neodymium 144.24 0.098 (3+,6) 0.181 HCP 0.366, 7.01 4 (soNd) 0.112(3+,8) 1.180 Neon (10Ne) 20.18 - - gas - - -Neptunium 237.05 0.101 (3+,6) 0.155 ortho - 20.25 -(93Np) Nick el 58.69 0.069 (2+,6) 0.125 FCC 0.3524 8.9 1 (2sNi) 0.056 (3+,6) Niobium 92.91 0.072 (3+,6) 0.143 BCC 0.3307 8.6 1 (41Nb) 0.079 (3+,8)

0.068 (4+,6) 0.048 (5+,4) 0.064 (5+,6) 0.074 (5+,8)

Nitrogen 14.007 0.016 (3+,6) 0.071 gas - - -(?N) 0.013 (5+,6) Nobelium 259.10 ? ? ? ? ? ? (102No) Osmium 190.23 0.063 (4+,6) 0.134 HCP 0.273, 22.59 2 (7s0s) 0.058 (5+,6) 0.439

0.055 (6+,6) 0.039 (8+,4)

Oxygen 16.00 0.140 (2-,6) 0.060 gas - - -(sO) 0.142 (2-,8) Palladium 106.42 0.086 (2+,6) 0.138 FCC 0.3890 12.02 1 (4sPd) 0.076 (3+,6)

0.062 (4+,6) Phosphorus 30.97 0.017 (5+,4) 0.109 ort ho - 1.83 -(1sP) 0.038 (5+,6) Platinum 195.08 0.080 (2+,6) 0.139 FCC 0.3924 21.4 1 haPt) 0.063 (4+,6) Plutonium 244.06 0.100 (3+,6) 0.159 manoei - 19.84 -(94Pu) Polonium 208.98 0.097 (4+,6) 0.140 SC 0.3366 9.32 1 (84Po) Potassium 39.10 0.137 (1+,4) 0.231 BCC 0.5344 0.86 1 (19K) 0.138(1+,6)

0.151 (1+,8)


Element Molar Ion ie Atomie Crystal lattice Solidor a/p mass radius/nm radius Structure parameter liquid /g·mor1 (charge, /nm cubic-hex density

coordi nation) a, c/nm /Mg·m-3

Praseodymium 140.91 0.099 (3+,6) 0.182 HCP 0.367, 6.77 4 (sgPr) 0.113(3+,8) 1.183 Promethium 144.91 0.097 (3+,6) 0.183 HCP 0.365, 7.22 4 (a1Pm) 0.109 (3+,8) 1.165 Protactinium 231.04 0.104 (3+,6) 0.156 tetra - 15.37 -

l91Paj •

Radium (88Ra) 226.03 0.148 (2+,8) 0.220 BCC 0.5148 "" 5.0 1 Radon (aaRn) 222.02 - - gas - - -Rhenium 186.21 0.063 (4+,6) 0.137 HCP 0.276, 21.02 2 (?sRe) 0.058 (5+,6) 0.446

0.055 (6+,6) 0.038 (7+,4) 0.053 (7+,6}

Rhodium 102.91 0.067 (3+,6) 0.134 FCC 0.3803 12.41 1 (4sRh) 0.060 (4+,6)

0.055 (5+,6) Roentgenium ca.280 ? ? ? ? ? (mRQ) Rubidium .244 BCC 0.5705 1.53 1 (37Rb)

Rutherfordium 261.11 ? ? ""23 ? (104Rf) Samarium 150.36 0.119 (2+,6) 0.180 rhomb - 7.52 -Ca2Sm) 0.127 (2+,8)

0.096 (3+,6) 0.108 (3+,8)

Scandium 44.96 0.075 (3+ ,6) CP 0.331, 2.99 2 Jz1Sc) 0.087 (3+,8) 0.527 Seaborgium 263.12 ? ? "'35 ? (106Sg) Selenium 78.96 0.198(2-,6) 0.117 ~ 0.437, 4.79 3 (34Se) 0.050 (4+,6) \--,ey 0.495

0.028 (6+,4) 0.042 (6+,6)

Silicon 28.09 0.026 (4+,4) 0.117 FCC 0.5431 2.33 2 (14Si) 0.040 (4+,6} Silver 107.87 0.100 (1+,4) 0.144 FCC 0.4086 10.5 1 (47Ag) 0.115 (1+,6f

0.128 (1+,8) Sodium 22.99 0.098 (1+,4) 0.186 BCC 0.4291 0.97 1 C11Na) 0.102 (1+,6)

0.118 (1+,8} Strontium 87.62 0.118 (2+,6) 0.215 FCC 0.6084 2.54 1 ba Sr) 0.126 (2+,8) Sulphur 32.06 0.184 (2-,6) 0.104 ort ho - 2.07 -(1aS) 0.037 (4+,6)

0.012 (6+,4) 0.029 (6+,6)

Appendices

Element Molar Atomie • Crystal

Tantalum (73Ta)

i Terbium l(!l§Tb) ·Thallium i (a1TI)

~orium goJh)

• Thulium • (a9Tm)

I Tin (soSn)

Tin (< 13 OC) Titanium (22Ti)

Tungsten (74W)

Uranium (e2U)

Vanadium (2N)

• Xenon (s.J(e) • Ytterbium • (?oYb)

Yttrium (~!lY)

radius i Structure lnm

ma ss i /g·mor1

180.95 0.143 BCC

HCP

127.60. 0.221 (2-,6) 0.137 0 066 (4+ 4)

' I o.o97 (4+.6)

0.043 (6+,4) 0.056 (6+,6)

158.93 0.092 (3+,6) 0.176 HCP 0.104 {3+,8)

204.38 0.150 {1+,6) 0.171 HCP 0.159 (1+,8) 0.075 (3+,4) 0.089 (3+,6) 0.098 (3+,8)

232.04 0.094 (4+,6) 0.180 FCC i 0,105 (4+,8)

168.93 0.088 (3+,6) 0.172 HCP 0.099 (3+,8)

I

118.69 0.055 (4+,4) 0.158 tetra 0.069 (4+,6) 0.081 (4+,8)

idem idem idem I FCC

47.88 i 0.086 (2+,6) 0.147 HCP 0.067 (3+,6) 0.042 (4+,4) 0.061 (4+,6) 0.074 (4+,8)

183.85 0.066 (4+,6) 0.137 BCC i 0.062 {5+,6) i

o.o42 {6+,4) I 0.060 (6+,6)

238.03 i 0.089 {4+,6) 0.138 ort ho . 0.100(4+,8)

50.94 I 0.079 (2+,6) I 0.132 BCC 0.064 {3+,6)

i 0.058 (4+,6) 0.072 (4+,8) 0.036 (5+,4) 0.054 (5+,6) •

131.29 - - gas 173.04 0.102 (2+,6) 0.194 FCC

0.114 (2+,8) 0.099 (3+,8) .....

88.91 0.090 (3+,6) 0.180 HCP 0.102 (3+,8)

I

I

I

I

I

Lattice parameter cubic-hex a, c/nm

0.3303

0.274, 0.439 0.446, 0 593

0.361, 0.570 0.346, 0.552

0.5084

0.354, 0.555

-

0.6489

0.295, 0.468

0.3165

-

0.3028

-0.5485

0.365, 0.573

205

Solidor alp • liquid density /M ·m-3

16.65

"'11.5 2

6.24 3

8.23 2

11.85 2 I

I

11.72 1

9.32 2

7.30

5.75 2

4.51 2

19.3 1

19.0 -

6.1 1

- -6.97 1

4.47 2


Element Molar Ion ie Atomie I Crystal Lattice Solidor alp ma ss radius/nm radius Structure parameter liquid /g·mol-1 (charge, lnm

1 cubio-hex denslty

coordination) a, clnm /Mg·m-3

Zinc 65.39 0.060 (2+,4) 0.133 HCP 0.266, 7.13 2 (JoZn) i 0.074 (2+,6) 0.495

0.090 (2+,8) . Zirconium 91.22 0.059 (4+,4) 0.158 HCP 0.323, 6.491 2 I (4oZr) 0.072 (4+,6) 0.515

0.084 (4+,8)

Appendix H Rubber elasticity

Appendices 207

For a material that is being deformed (L increases due to force F) we can write for the change in interna I energy: dU = oQ oW = TdS FdL.

This can also be written as: F = ( ~~) T( :~) The first term is familiar: it is the change in internal energy due to stretching of bonds between atoms. This leads to the Hookean behaviour we saw in Chapter 3.

The second term is a different form of elasticity, which is usually called rubber elasticity. In order to explain this we must first look at the entropy of a rubber.

Fora polymer chain that is folded randomly, we can use the random-coil model to delermine the entropy: there is a distribution tunetion W(x,y,z) (see Appendix N), and entropy is nothing more than this distri bution written in different form:

S =ka In W= Cs- kabs2r/ =Cs- kabs2(~ +I+ z2) [J/K]

When we now deform the material in e.g. the z direction, we get a new dimension z' = "Az. With the usual assumption that the volume is constant (not always true!), we use: V= xyz = xy'z'.

Then, there is equal contraction along the x and y directions: x'= li and y' = } .

Then we can write for the change of entropy of a single polymer chain during this deformation:

él.Sse = -kabs2{~( i -1) + 1( Î 1) + z2("A 2 - 1)} [J/K]

The material of course contains more polymer chains: Ne per m3. The density of the chains (dNe) in a certain volume (dxdydz) is the same as finding one chain end in a. certain volume, so we can use the familiar distribution tunetion again:

[ b

2 ) b' 2 dNe = NeW(x,y,z)dxdydz =Ne 1C 3~2 e- ,r, dxdydz

The total change of the entropy for all chains in 1 m3 is now the inlegral of ll.SscdNe. Fortunately this is a standard inlegral that can be calculated quite easily:

This formula behaves as expected: forA. > 1 and "A < 1 we find él.S < 0, so stretching and campressing of the rubber network is less favourable as far as entropy is concerned.


When we have a piece of material with an initia! length Lo and initia! cross section Ao, we get as total entropy change for the whole chunk of polymer:

(

À2 + 2 3] AStot = -NcksAoLo ; [ J/K]

How do we translate this entropy change in the familiar terms of stress and strain?

For the stress we can write cr = F , and in a rubber (with very weak farces between Ao

the chains) we can write as shown above: F ""'-< ~~) Since À = L this gives for the stress:

Lo

Often it is written that Ncks T = G, the shear modulus of the rubber. Why this is done can be seen by the following arguments: first, there is a conneetion between the elangation factor À and the strain e:

L À=-= 1 + e

Lo

For small values of E (~: « 1) we can then approximately write:

1 = 1 2e

Then the relation between cr and E becomes:

cr G(1 + E- 1 + 2~::) = 3Ge

So: G ~. This is exactly what is expected fora rubber with a Poisson ratio (v) of 0.5

(remember: constant volume was assumed).

For rubber elasticity we can then write:

Appendix I Polymer structures

Appendices

Name Abbrevlatlon

Poly( ethene )/poly( ethylene) PE

Poly(propene )/poly(propylene) PP

Poly(styrene) PS

Poly(vinyl chloride) PVC

Poly(ethyl acrylate) PEA

Poly(methyl methacrylate)/Piexiglass PMMA

Poly(tetrafluorethe ne )/Teflon PTF E

Poly{butadiene)

Poly(isoprene) PI

Poly( dimethylsiloxane) PDMS

Poly( ethylene terephthalate) PET

Repeating unit

H -CH-C-

2 I ,c,

o' OEt

-CH-2 -C=CH-CH2-I CH3

209

2 2 -...::::

0 -O-CH-CH-0~

I~ (

Poly(ethylene oxide)/poly(ethylene glycol) PEO/PEG -CH2CH20-0

Poly(lactic acid) PLA


Name Abbreviotion Repeoting unit

0

Poly(glycolic acid) PGA -O-CH~

H Poly(vinyl acetate) PVAc -CH2-y-

OÎ(Me

0

Poly( acrylanitri Ie) PAN H

-CH-C-2 I

CN

Poly(vinyl alcohol) PVA H -CH-C-

2 I OH

Poly(formaldehyde) -CH20-

0 Nylon-6 -~-(CH2)s----<

0 Nylon-6,6 -N-(CH2)6-N-----< ----(.

H H (CH2)4 0

--of.o-01(0-Poly( carbonate) PC CH3 0

Poly(isobutene) yH3

-CH-C-2 I

CH3

Remember: Et = CH3CHr, Me = CH3

Appendix l Phase rule

Appendices 211

The phase rule can be derived quite easily. Suppose we have C compounds, distributed over P phases. How many things can we choose in principle? For each phase, we must specity the pressure, the temperature and the chemica! composition. lf there is equilibrium, the temperature and pressure are the same for all phases, so this gives only 2 things to specity (pandT). In each phase, we specity the chemica! composition as mole fractions (x;), which must always obey I:x; = 1. Therefore, we only need C- 1 mole fractions to specity the chemica I composition of each single phase. For P phases, we therefore need P(C- 1) mole fractions. So we could in principle choose 2 + P( C - 1 ) varia bles.

However, if there is equilibrium the chemica! compositions are not independent in all phases. lt can be shown from thermodynamics that each compound has a socalled chemica! potential (Jli), which must be the same in all phases if there is to be equilibrium. For compound i in phases a., J3, y, 3, ... we must therefore have:

For P phases, this means that there are P- 1 relations that must hold for each component i in all phases J3, y, o, ... :

IJ-;[3 = Jli" ~-t? = 1-li" 1-llö = 1-li" etc. for all remaining phases

lf we do this for all C components, we get C(P- 1) relations that must hold, and that limit the choice of the variables found in the first paragraph.

So we are left with a number of choices that we can freely make: the total number of variables minus the relations necessary for equilibrium. Or, if we call this the degrees of freedom:

:F = [2 + P( C- 1)] - C(P- 1) = C + 2 - P

This is the phase rule.


Appendix K Lever rule

The lever rule tells us how to derive the physical composition of a two-phase mixture (i.e. how much of the phases is there?) from the chemica/ compositions (i.e. how much of each compound is there in each phase?). How does this work?

Suppose we have mo kg of a mixture of compounds A and B. lf the average chemica! composition of this mixture is xo (as mass fraction of A), then there is mA= xomo kg of A and ms = (1-xo)mo kg of B.

When this mixture separates into two phases (1 and 2), we get the masses m, and m2 of the phases, with of course m1 + m2 = m0. We are now interestad in the mass fractions f1 = m1/mo and fz = mz/mo. These describe the physical composition. Of course, f, + f2 = 1.

Each phase will have its own chemica/ composition: phase 1 wilt have a mass fraction of A of x,, and phase 2 will have a composition Xz. Then in phase 1 there will be x1m1 kg of A, and in phase 2 there will be xzmz kg of A. In total, the amount of A cannotchange,so:

When we divide by m0, we can also write this as:

When we now use the fact that fz = 1 - f1, we can isolate t, to get:

Again using fz = 1 - f1, it is now easy to get:

These formulae consitute the lever rule.

lf you are lost in all the symbols, perhaps this simple example will help:

mo 5 Xo = 3/5

m, = 2 ,, = 2/5 x,= 1/2

mz=3 fz = 3/5 Xz = 2/3

Appendices 213

Appendix L Packing in ionic solids

When cations (+) and anions (-)are combinedintoa crystal, their sizes will generally not be the same, and this delermines (roughly) the kind of packing or crystal structure you get. We will use the ratio r+/r_: since cations are usually smaller than anions, this is in general smaller than 1. We willlook what happens when the cation becomes ever smaller. An easy tooi is then the coordination number: the number of directly neighbouring ions (Ne).

When r+lr- < 1, the first situation we get is Ne = 8: the so-called cubic arrangement seen in the CsCI structure, where the 8 anions at the corners of a cubic unit cell surround one cation in the centre of the cube (see Figure 4.7). The limit of this situation is reached when the cation becomes so small that the anions start touching along the edges of the cube: when the cation in the centre becomes even smaller, not all 8 anions fit around it anymore, so we switch to Ne = 6. We can calculate the radii at which this transition happens: when all the ions touch, we have a= 2r_ for the edge of the cube, and a.V3 = 2r+ + 2c for the body diagonal (from one corner to the opposite; as for BCC). This yields: r+/r_ = .V3 1 = 0.732. Therefore, we will (roughly) get a CsCI structure for 0.732 < r+/r_ < 1.

Not convineed about the .V3 in the body diagonal? First go diagonally to another corner along the side; then use Pythagoras: when the side is a, the length of the side diagonal is .V(a2 + a2

) = a.V2. Then go along the side and do Pythagoras with this side and the side diagonal: this will give you the length of the body diagonal as .V{(a.V2)2 + a2

} = .V{2a2 + a2} = .V{3a2

} = a.V3. QED

When r+/r_ < 0.732, we get Ne= 6: the octahedral arrangement as seen in the rock salt (NaCI) structure, where the 6 anions in the fa ces of the cubic unit cell surround a cation in the centre of the cube (see Figure 4.11 ).

We reach the limit of this arrangement when anions start touching along side diagonals (the same as a "normal" FCC): this is seen clearly in e.g. the bottorn plane of the NaCI structure. Forsmaller cations we switch to Ne = 4.

The radii at which this happens follow trom the situation where all ions just touch along the side diagonal: we then have a.V2 = 4r_, and the side of the cubic unit cell hereis a= 2r+ + 2r_. This gives: r+/r_ = .V2 -1 = 0.414. Therefore, the NaCI structure will be formed when 0.414 < r+/r_ < 0.732.

When r+/r_ < 0.414, we get Ne= 4: the tetrahadral arrangement as seen in the zinc blende structure, where anions at the corners and in the faces of the cubic unit cell surround cations inside the cube. The cation is in fact in the centre of a smaller cube with side a/2 and 4 anions around it at the corners (Figure 4.1 0).

Here the limit is reached when the anions start touching along the side diagonal of the cube (as for the one-atom-per-point FCC). Then we have for the side diagonal: a.V2 = 2r_, and for the body diagonal of the smaller cube: (a/2).V3 = 2r+ + 2L Th is yields: rJr_ = .V6- 1 = 0.225. We will therefore get the zinc blende structure when 0.225 < r+/r_ < 0.414.

When r+/r_ < 0.225, we get Ne= 3 (a triangular arrangement), and for r+/r_ < 0.155 we get Ne= 2 (a linear arrangement). We will not look at these cases, as they are not relevant for us.


Appendix M The Orowan model for brittie fracture

The Orowan model for fracture is a simple model that gives a theoretica I (and incorrect) estimate of the tensile stress of a crystalline material.

We assume a perfectly crystalline material, and want to separate two layers of atoms, which are perpendicular to the stress. Normally the atoms are situated at a distance Ro, and this has to be stretched befare the bond can be braken.

cr

Ro cr

We will estimate the maximum stress with a simple model, where the stress first increases to a maximum, and then decreasas until the atoms are free of each other .

. n(R- R0 ) a = a max Sin-----''-

R1

The stress now has a maximum of Gmax at a distance R = Ro + R1/2, and fracture is complete at a distance R = Ro + R1.

In order todetermine the maximum stress (or tensile strength), we use the following technique.

For R "" Ra we can assume elastic behaviour:

So for the initia! slope we can write:

da E dR Ra

From the sine-model, we can also calculate the slope:

Appendices 215

Now, for R = Ro this slope is:

From these two equations for the slope, it follows that the maximum stress is related to an elastic property, viz. Young's modulus:

O'max (i)

In order to delermine the value of R1. we havetolook at the energy that is necessary to break the bonds on a surface of 1 m2

. This is given by the integral of the stress, using the sine-model for the stress:

Ro+R, 2 R !:.E = J crdR = O'max 1

R 1t 0

Th is energy must now be equal to the energy necessary to create two new surfaces of 1 m2

: t:.E = 2y. From these equations it follows that:

R TT.Y 1 -- (ii)

O'max

When we combine this with equation (i) that we have derived above, we get for the maximum stress, which is the tensile strength:

O'max {li vRc:

We can also use equations (i) and (ii) to derive a formula for the surface energy:

Experimentally we often find that R1 ~ Ro, so y ~ ERo/10, and wethen find for the tensile strength:


Appendix N The random-coil model in three dimensions

In three dimensions, the random walk model is more complex, since we can move in more directions: every step can be at a different angle to the previous one. We will

write the step i as vector ~ with length L, so the total walk will be:

R =IJ I

Now we again use the square, but in this case via the dot product:

RL- = R · R = (~ ~ )·(~ ~) = ~( ~ · ~) + 2~ ~ • ~ (the last term with i < J)

On averaging over all possible directions, we get:

<RL-> = ~< ~ · ~ > + 2~< ~ · ~ > = NL 2 + 2~1 Î; I I~ I <cos 0>

Since <cos 0> = 0 (all values are present both positive and negative in the interval 0 to 2n, or the inlegral over (0,2:rc) of cos 0 is 0), we are again left with:

<Ff-> = NL2

This can also beseen with a more complicated model: the Gaussian chain. lf the polymer chain consists of ns segments, each with length Ls, then the distribution of the second chain end location (at a distance re from the origin) is given by a Gaussian distribution:

b3 2 2

W( ) s -b,r, [m-3] x, y, z = -:ïi2 e 1C

with:

The chance to find the second chain end at a distance re in any possible direction, so on a spherical surface of radius re, is then given by:

W(rc)=4:rcr;w(x,y,z) [m-1]

The end-to-end distance re will of course be different for different polymer chains, and at different times. In order to give an average size of the randomly coiled polymer chain, one usually works with the root-mean-square value of the end-to-end distance re. This average can also be compared easily to experimental results. lt can be shown that this average value is:

00

(rn = fr~W(re)dre = n8L~ [m2

]

0

Appendices 217

Appendix 0 Answers to Problems

2.1.a. 2.0011 m b. 3.6m

2.2. f; = 0.004

2.3.a. AL= 0.31 mm b. Ad= -0.0092 mm

2.4.a. 5.8 km b. .-.. = 0.00217; top Cë...ftf/".0-Y:-:; oj 0 o "-111 c. botlom

2.5.a. E = 138 GPa b. ay = 345 MPa c. crr = 1034.5 MPa d. 0.43 MJ/m3

e. 110.8 MJ/m3

f. 15%

2.6.a. E = 170 GPa b. ar= 410 MPa c. 85 kJ/m3 (for r:v = 0.001) d. 18.3%

2.7.a. E = 353 GPa b. crr = 1250,MPa c. 46% d. crt = 1650 MPa (via: crt = cr(1 + e)) or cr1 = 2040 MPa (via: cr1 = crAo/A)

2.8.a. et = 0.3 b. e = 0.35 c. crt = 913 MPa, a= 676 MPa

2.9.a. et = 0.237 b. e1 = 0.20: e = 0.22, a= 471.3 MPa; Gt = 0.237: e = 0.267, a= 473.6 MPa

2.11.a. do = 1.99 mm b. F = 1766 N

3.1.a. Ro = (2B/A) 116

b. B/A = 1.22x1 0-58 m6

c. U(Ro) = -AI(2Ro6)

3.2.a. LiF (690 GPa) b. LiF: -0.57 MJ/mol; NaBr: -0.41 MJ/mol


3.3.a. both 0.86x1 o-11 N, but with different signs b. compression: -1.1 nN; tension: +0.68 nN

3.4.a. Uo = D(1 Ro/p}exp(-Rolp) b. Uo = (p/Ro -1)(C/Ro)

3.5. n = ~ t , :t 5

4.1.a. 0.2633 nm b. 0.1798 nm

W ;l.~- I ~~~~\-4.2.a. FCC r- • b. BCC Wl t\- \ ~ I p~'r

4.3. 6.62x10-29 m3

4.4. 6.04x1 028 atoms/m3

4.5. 3.7%

4.6. 3.45%

4.7. 1.85 Mg/m3

4.8. 3.37 Mg/m3; 8 atoms per cell

4.9.a. 4 each b. FCC

4.10. 1: [1 41]; 2: [121]; 3: [120]

4.11. a:(212);b:(014);c:(101)

4.12. f,h,andi

4.13.a. (113) b. x-axis: (3,0,0); y-axis: (0,3,0); z-axis: (0,0, 1)

c. [1 T OJ En [ T I 0]

4.14. p = 2.34 Mg/m3

4.15.a. r = 0.125 nm b. 2 atoms/cell

4.16. Any vector with h + 4k + 2/ = 0

5.1. 28 kN

5.2.a. 0. 73 mJ b. 94.1 J

Appendices

5.3.a. YLS = 0.66 J/m2

b. WcHg = 0.94 J/m2, Wcglass = 0.60 J/m2

' 2 ' 2 c. Wa = 0.11 J/m, S = -0.83 J/m

5.4.a. 500 MPa b. 2.2 mm

5.5. 1.8 mm

5.6. cr* = 16.2 MPa

5.7. Fracture occurs

5.8.a. p = 0.4 nm b. cr* = 1.48 GPa

5.9.a. cr* = 590 MPa b. y = 360 J/m2

5.10.a. cr = 567.7 MPa b. ey = 0.0015

5.11. 11.7%

5.12. [120]

6.2. DP = 1240

6.3. 45.6 kg/mol

6.4. DP = 330

6.5.a. lsoprene: 0.443; butadiene: 0.557 b. 5.31 g

6.6. Mn = 24.15 kg/mol; Mw = 24.68 kg/mol

6.7.a. Mn = 33.04 kg/mol b. Mw = 36.24 kg/mol c. 786.7 d. 862.9 e. D = 1.097

6.8.a. 0.802 b. 0.420 c. Worse packing

6.9. 0.091

219


6.1 O.a. Smaller side groups: more flexible polymer chain b. Smaller side groups: more flexîble polymer chain c. Dipolar interactions and H-bonds: less flexible d. Double substitution: less flexible

6.11. <r/>112 = 3.4 nm; R9 = 1.39 nm

6.12. 3.05 mol

7 .1.a. Crystallinity b. Crosslinks c. T emperature d. Crosslink number

7.2. 83.3 days

7.3.a. 11 = 6.06x1011 Pa·s b. 8 = 0.01

7.4.a. TRhK = 0.25 c. TK = 0.2 s; tR = 0.05 s; tan(&} = 0.6, 0.75, 0.69 at 5, 10, 15 rad/s, respectively.

7.5. À= 1.5: cr = 0.53 MPa; À= 0.8: cr = -0.38 MPa

7.6. c* = 0.61 mm

7.7. a= 55 !J.m

7.8. crr = 44 MPa

7.9.a. w = 1.5 MJ/m3

8.1. ro = 10.43 mm

8.2. Cold-work to 36 %CW: d = 12.8 mm; anneal; cold-work to 22 %CW.

8.3. 2.87 mm to 0.34 mm

8.4. d = 0.0147 mm

8.5.a. 4750 g b. ca. 64 wt% sugar

8.6.b. ca. 6.5 wt% salt

Appendices 221

8.7.a. 94 mol% Sn, 6 mol% Pb b. :f = 2: above 220 oe; :f = 1: between 220 and 183 oe; :f = 0: exactly at 183

oe; :f = 1: below 183 oe

c. 90 wt% Sn: J3-particles in layered eutectic matrix; 10 wt% Sn: p-particles in amatrix

8.8. 2800 oe: only L, XL = 50 wt% Re; 2452 oe: L + p, XL = 43 wt% Re, Xp = 56 wt% Re, 46 wt% L, 54 wt% p; 2448 oe: a + J3, Xa = 48 wt% Re, XJ> = 56 wt% Re, 75 wt% a., 25 wt% J3; 1000 oe: a+ J3, Xa = 40 wt% Re, x~ = 59 wt% Re, 47 wt% a., 53 wt% J3

8.9.a. Just below 114 7 oe: y + Fe3e. Xy = 2.14 wt% e. XFe3C = 6.67 wt% e. 70 wt% y, 30wt% Fe3e; 900 oe: y + Fe3e. Xy = 1.27 wt% e, XFe3C = 6.67 wt% e. 59 wt% y, 41 wt% Fe3e; just below 727 oe: a+ Fe3e. Xa = 0.022 wt% e, XFe3C = 6.67 wt% e, 48 wt% a., 52 wt% Fe3e

8.10. 36.3 s

9.1.a. 32GPa b. 4.4 GPa

9.2. 7%

9.3. Em = 3.15 GPa; Et= 72.1 GPa

9.4. 33.3 MPa

9.5.a. 0.125 mm b. Vt = 0.799 c. Wr = 62.4 kJ/m2

d. Ec = 61.3 GPa e. K1c = 87.5 MPa·m112

9.6.a. crT = 10.8 MPa b. E1 = 2.1 MPa; E2 = 34.6 MPa d. 1.73 MJ/m3

9.7.a. lsostrain: v, = 0.31; isostress: Vt = 0.48 b. 1:tm = 19.2 MPa


Index A adhesion, 69 aggregation state, 132 amorphous, 54, 92, 108 anisotropic, 33, 47, 67 annealing, 130 atactic, 83 austenite, 140

B bainite, 143 BCC,43 binary phase diagram, 132 bioceramic, 5, 36 biocomposites, 158 biological material, 5 biomaterial, 5 biomineralization, 36 biopolymer, 6 block copolymer, 82, 145 body-centred cubic, 42 bond energy, 28 Bravais lattice, 40 brittle, 17 brittie fracture, 16, 68 bulk, 6 bulk composition, 69

c cancellous bone, 164 cartilage, 160 cast iron, 140 CCT -diagram, 144 cellular material, 5 cellulose, 99 cementite, 140 ceramic, 5, 36, 75 chemica! bonds, 6 chemica! composition, 132 close-packed structure, 43,45 cohesion, 69 coil,84 cold working, 129

collagen,96 compact bone, 164 component, 136 composite, 5, 152 compression, 8 connective tissue, 158 contact angle, 68, 88 copolymer, 82 Coulomb, 6, 28 covalent bond, 6 covalent network, 6 crack, 71 creep, 109 critica! crack length, 74 critica! shear stress, 62 critica! stress, 7 4 crosslink, 84 crystal, 38 crystalline, 36 crystal system, 39 CsCI-structure, 42 cubic structure, 40

D defect, 63 deformation angle, 10 degree of freedom, 136 degree of polymerization, 86 dentine, 166 dermis, 161 dewetting, 69 diamond cubic structure, 44 dipole-dipole interaction, 6 dislocation, 64 dislocation mobility, 66, 128 dispersion hardening, 129 dot product, 50 drop formation, 69 ductile, 17 ductile fracture, 16 ductility, 14 dynamic deformation, 114

Index --------------------------------------223

E edge dislocation, 64 elastic component of modulus, 115 elastic deformation, 10 elastic energy, 12 elastin, 98 elastomer, 85 electrastatic bond, 6, 28 elangation factor, 9 enamel, 166 engineering strain, 20 engineering stress, 18 entanglements, 84 entropy, 119 epidermis, 161 equilibrium distance, 29 eutectic, 136 eutectic structure, 138

F face-centred cubic, 42 FCC,43 ferrite, 140 fibre, 152, 157 film formation, 69 flat bone, 164 flexible, 17 flexural stiffness, 10 force, 8, 31 fracture, 16,68,121 fracture toughness, 74 free volume, 90

G GAG, 99 gel, 99 glass, 36, 54, 78 glass (transition) temperature, 55, 90 glass transition, 55, 90 glycosaminoglycan, 99 grain, 47 grain boundary, 47 grain size, 131, 143 Griffith model, 73

H hardness, 17 Haversian bone, 164

HCP,45 hexagonal structure, 40 hexagonal close packed structure 45 hierarchical material, 5 ' Hooke's law, 10 hydragen bond, 6 hysteresis, 13, 115

internal friction, 115 interatomie distance, 28 ion, 6, 29 isostrain model, 152 isostress model, 152 isotactic, 83 isotropic, 11, 33, 47, 67

K keratin, 97

L lamellar,89 lamellar bone, 164 laminar, 152 laminar bone, 164 lamp brush model, 158 lattice, 38 lattice parameter, 39, 40 lattice point, 39, 42 length scale, 4, 78, 95 lever rule, 134, 211 line defect, 64 linear elastic solid, 110 liquidus, 133 loss component of modulus, 115

M martensite, 143 macromolecule, 6, 82 macroscale, 4, 78 maximum strain, 14 Maxwell model, 110 machanical properties, 4, 17, 78, 105 melting temperature, 55, 90 mesoglea, 160 mesoscale, 4, 78, 95 metal, 5, 36, 76, 128 metallic bond, 6


microscale, 4, 78, 95 microstructure, 65 Milier indices, 49 molecule, 6 monolithic material, 5 morphology, 65

N nacre, 163 necking, 18 network, 84 number-averaged molar mass, 86

0 optimal fibre length, 155 Orowan model, 70, 213 osteon, 164

p particulate, 152 pearlite, 141 phase, 128, 132 phase angle, 114 phase diagram, 132 phase rule, 136, 210 physical composition, 134 planar defect, 65 plastic deformation, 13 plasticizer, 122 point defect, 64 Poisson ratio, 11 polycondensation, 82 polycrystalline, 4 7 polydispersity, 87 polymer, 5, 82, 145 polymer crystals, 89, 107 polymerization, 82 polymorphic, 47 polysaccharide, 5, 96 PPL, 160 precipitation hardening, 128 protein, 5, 96 protein-polysaccharide complex, 158 proteoglycan, 158

R radical polymerization, 82 radius of curvature, 72

radius of gyration, 93 random-coil model, 92, 215 recovery, 15 resilience, 12 resin, 84 Reuss model, 152 rock salt structure, 44 root-mean-square value, 93 rubber, 85 rubber elasticity, 119, 206

s salt, 6, 29 SC,42 screw dislocation, 64 secondary interactions, 6 shear modulus, 1 0 shear stress, 9 shearing, 9 silk, 98 simple cubic, 42 simple material, 5 single crystal, 47, 61 skin, 161 slip system, 61 solid solution, 128 solidus, 133 spherulite, 89 spreading coefficient, 69 steel, 140 stiff, 17 stiffness, 1 0, 1 06 starage component of modulus, 115 strain, 9 strain hardening, 14, 129 stress, 8 stress concentration, 72 stress intensity, 7 4 stress relaxation, 109 strong, 17 surface composition, 69 surface tension, 68, 88 syndiotactic, 83

T tendon, 162 tensile strength, 14 tension, 8 thermoharder, 84

thermoplast, 84 tie line, 133 tooth, 166 tough, 17 toughness, 16, 74, 156 true strain, 20, 185 true stress, 1 8, 1 85 TTT-diagram, 142

u unit cell, 38

V van der Waals attraction, 6 visco-elastic, 104, 109 Voigt model, 110, 152 Voigt-Kelvin model, 110

w weak, 17 weight-averaged molar mass, 86 wetting, 69 whisker, 70 work hardening, 14, 129 workof fracture, 76 woven bone, 164

y yield, 13, 60 yield strain, 13 yield strength, 13 yield stress, 13 Young's modulus, 10

z zinc blende structure, 44

Index 225

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Materials science for biomedical engineeringEvelinda Baerends. Of course, all errors that are undoubtedly still in this book are my responsibility. Second edition Marcel van Genderen